🧠:Okay, let's try to figure out how to find the volume of this pyramid. The problem says it's a right triangle base with hypotenuse c and an acute angle of 30 degrees. The lateral edges are inclined at 45 degrees to the base plane. Hmm, volume of a pyramid is usually (1/3)*base area*height, right? So I need to find the base area and the height of the pyramid.First, let me visualize the base. It's a right triangle with a hypotenuse of length c and one acute angle is 30 degrees. So in a right triangle, the angles are 90, 30, and 60 degrees. The sides opposite these angles should be in the ratio 1 : √3 : 2, right? Since the hypotenuse is c, which corresponds to the 2 in the ratio. So the sides opposite 30 degrees would be c/2, and the side opposite 60 degrees would be (c√3)/2. Let me confirm that.In a 30-60-90 triangle, the sides are in the ratio 1 : √3 : 2, with the hypotenuse being twice the shortest side. So if hypotenuse is c, the shortest side (opposite 30 degrees) is c/2, and the other leg is (c/2)*√3. So yes, the base right triangle has legs of length c/2 and (c√3)/2. Therefore, the area of the base would be (1/2)*(c/2)*(c√3/2) = (1/2)*(c²√3)/4 = c²√3/8. Okay, that's the base area.Now, the height of the pyramid. The problem says the lateral edges are inclined at 45 degrees to the plane of the base. Hmm, lateral edges... In a pyramid, the lateral edges are the edges from the base vertices to the apex. So each of these edges forms a 45-degree angle with the base plane. Wait, so if we consider one of these lateral edges, its projection onto the base plane would be from the apex's projection to the base vertex. Since the apex is directly above the base's centroid? Wait, no, in a regular pyramid, but this base is a right triangle. Wait, but it's not specified if the pyramid is regular or not. Wait, actually, the term "pyramid" with a triangular base is just a tetrahedron, but with a specific base. However, the problem mentions "lateral edges" inclined at 45 degrees. Let me think.In a pyramid, the lateral edges are the edges connecting the apex to each of the base vertices. Since the base is a right triangle, the pyramid is a tetrahedron with a right triangle base. The lateral edges (the three edges from apex to each vertex of the base) are each inclined at 45 degrees to the base plane. That means each of these edges makes a 45-degree angle with the base.So, if we can find the height of the pyramid, that's the perpendicular distance from the apex to the base. Then, using the angle of inclination of the lateral edges, which is 45 degrees, we can relate the height to the horizontal distance from the apex's projection to each base vertex.Wait, but each lateral edge is inclined at 45 degrees. So for each lateral edge, the angle between the edge and the base is 45 degrees. Therefore, the height of the pyramid h and the horizontal distance from the projection of the apex to each base vertex would form a right triangle with the lateral edge as the hypotenuse, right? Since the angle between the lateral edge and the base is 45 degrees, then tan(45) = h / d, where d is the horizontal distance. But tan(45) is 1, so h = d.But here's the problem: the apex's projection onto the base (let's call it point O) must be equidistant from all three vertices of the base? Wait, in a general pyramid, unless it's a right pyramid, the apex isn't necessarily over the centroid or any specific point. But in this case, if all lateral edges are inclined at 45 degrees, that imposes a condition on the position of the apex.Wait, if all three lateral edges make a 45-degree angle with the base, then their projections onto the base (which would be the distances from O to each vertex) must equal h, since h = d (because tan(45) = 1). Therefore, the apex is directly above a point O in the base such that the distance from O to each vertex of the base is equal to h. But in a triangle, the set of points equidistant from all three vertices is the circumcenter. So in this case, point O must be the circumcenter of the base triangle, and the distance from circumcenter to each vertex is the circumradius, which would be equal to h.Therefore, h is equal to the circumradius of the base triangle. So if I can compute the circumradius of the right triangle base, that would give me the height h.For a right triangle, the circumradius is half the hypotenuse. Wait, yes! In a right triangle, the hypotenuse is the diameter of the circumcircle, so the circumradius is half the hypotenuse. Therefore, the circumradius R = c / 2. Therefore, h = R = c / 2. Wait, but if h = d, and d here is the circumradius, then h = c / 2.Therefore, the height of the pyramid is c / 2. Then, the volume would be (1/3)*base area*h = (1/3)*(c²√3 / 8)*(c / 2) = (1/3)*(c³√3)/16 = c³√3 / 48.Wait, but let me verify this again because sometimes I might confuse the centroid and circumcenter. For a right triangle, the circumcenter is indeed at the midpoint of the hypotenuse. So the distance from the circumcenter to each vertex is half the hypotenuse. So yes, if the apex is directly above the circumcenter, then the horizontal distance from the apex's projection (which is the circumcenter) to each vertex is c/2. Therefore, since the angle of the lateral edge is 45 degrees, the height h equals that horizontal distance, so h = c / 2.Therefore, volume is (1/3)*(base area)*h = (1/3)*(c²√3/8)*(c/2) = (c³√3)/(48). So the volume is c³√3 divided by 48.But let me check another way. Suppose we take coordinates. Let's place the right triangle in the coordinate system with the right angle at the origin, one leg along the x-axis, and the other along the y-axis. The hypotenuse would be from (a, 0) to (0, b), where a and b are the legs. Given that the hypotenuse is c, so a² + b² = c². The acute angle is 30 degrees, so let's say the angle at the origin is 90 degrees, then the other angles are 30 and 60. Wait, the problem says "a right triangle with a hypotenuse equal to c and an acute angle of 30°". So one of the non-right angles is 30°, which makes the other 60°. Therefore, the legs are opposite these angles.So the side opposite 30° is the shorter leg, let's say length a = c/2, and the side opposite 60° is b = (c√3)/2. Then, the base triangle has vertices at (0,0,0), (c/2, 0, 0), and (0, (c√3)/2, 0). Wait, but hypotenuse length would be sqrt((c/2)^2 + ((c√3)/2)^2) = sqrt(c²/4 + 3c²/4) = sqrt(c²) = c, which checks out.Now, the apex of the pyramid is at some point (x, y, h), where (x, y, 0) is the projection onto the base. The lateral edges from the apex to each vertex of the base are inclined at 45°, so the angle between the lateral edge and the base is 45°. The lateral edges are from (x, y, h) to (0,0,0), (c/2, 0, 0), and (0, (c√3)/2, 0).The angle between each lateral edge and the base is 45°, which is the angle between the lateral edge and its projection onto the base. The projection of the lateral edge onto the base is the vector from (x, y, 0) to the vertex. The length of the projection is the distance from (x, y, 0) to the vertex, which is sqrt((x - xi)^2 + (y - yi)^2) for each vertex (xi, yi, 0). The length of the lateral edge is sqrt((x - xi)^2 + (y - yi)^2 + h²). The angle between the lateral edge and the base is given by the angle between the edge and its projection, which is 45°, so:cos(45°) = adjacent / hypotenuse = (projection length) / (edge length)But wait, actually, the angle between the lateral edge and the base is measured between the edge and the plane. The angle between a line and a plane is defined as the complement of the angle between the line and the normal to the plane. Wait, no, actually, the angle between a line and a plane is the angle between the line and its projection onto the plane. So yes, that angle would be 45°, which means that the angle between the lateral edge and its projection is 45°, so:tan(45°) = h / (projection length)But tan(theta) = opposite / adjacent, where theta is the angle between the edge and the projection. So if the angle is 45°, then tan(45) = h / d = 1, so h = d, where d is the projection length (distance from (x, y, 0) to the vertex). Therefore, for each lateral edge, h must equal the distance from (x, y, 0) to each vertex. Therefore:sqrt((x - 0)^2 + (y - 0)^2) = hsqrt((x - c/2)^2 + (y - 0)^2) = hsqrt((x - 0)^2 + (y - (c√3)/2)^2) = hSo these three equations:1. x² + y² = h²2. (x - c/2)² + y² = h²3. x² + (y - (c√3)/2)^2 = h²Subtracting equation 1 from equation 2:(x - c/2)² + y² - x² - y² = 0Expanding (x - c/2)^2: x² - c x + (c²)/4 - x² = -c x + c²/4 = 0So -c x + c² /4 = 0 → x = c/4Similarly, subtract equation 1 from equation 3:x² + (y - (c√3)/2)^2 - x² - y² = 0Expanding: y² - c√3 y + (3 c²)/4 - y² = -c√3 y + (3 c²)/4 = 0So -c√3 y + 3c² /4 = 0 → y = (3c² /4) / (c√3) = (3c) / (4√3) = (c√3)/4 after rationalizing.Therefore, the projection of the apex is at (c/4, (c√3)/4, 0). Then, the distance from this point to any vertex is h.Let's compute h using equation 1: x² + y² = h²x = c/4, y = c√3 /4So h² = (c²)/16 + (3 c²)/16 = (4 c²)/16 = c² /4Therefore, h = c / 2, which matches our earlier conclusion. Therefore, the height is c/2.Therefore, the volume is (1/3)*base area*h = (1/3)*(c²√3 /8)*(c/2) = (c³√3)/48.So the volume is c³√3 divided by 48.Wait, let me check the calculation again. Base area was (1/2)*(c/2)*(c√3/2) = (c²√3)/8. Then height is c/2. So (1/3)*(c²√3/8)*(c/2) = (c³√3)/(48). Yes, that's correct.Therefore, the final answer should be boxed{dfrac{c^3 sqrt{3}}{48}}.Final AnswerThe volume of the pyramid is boxed{dfrac{c^3 sqrt{3}}{48}}.

✅:To find the volume of the pyramid with a right triangle base and lateral edges inclined at (45^circ) to the base plane, we follow these steps:1. Determine the dimensions of the base triangle: - The base is a right triangle with a hypotenuse (c) and an acute angle of (30^circ). - The sides of the triangle are in the ratio (1 : sqrt{3} : 2). For hypotenuse (c), the legs are (frac{c}{2}) (opposite (30^circ)) and (frac{csqrt{3}}{2}) (opposite (60^circ)).2. Calculate the area of the base: - The area (A) of the right triangle is: [ A = frac{1}{2} times frac{c}{2} times frac{csqrt{3}}{2} = frac{c^2 sqrt{3}}{8} ]3. Determine the height of the pyramid: - The lateral edges are inclined at (45^circ) to the base plane. This implies the height (h) of the pyramid is equal to the horizontal distance from the projection of the apex to each vertex of the base. - For a right triangle, the circumradius (distance from the circumcenter to each vertex) is half the hypotenuse. Thus, the height (h) of the pyramid is: [ h = frac{c}{2} ]4. Calculate the volume of the pyramid: - The volume (V) of a pyramid is given by (frac{1}{3} times text{base area} times text{height}): [ V = frac{1}{3} times frac{c^2 sqrt{3}}{8} times frac{c}{2} = frac{c^3 sqrt{3}}{48} ]Thus, the volume of the pyramid is (boxed{dfrac{c^3 sqrt{3}}{48}}).

🔑:1. Let's start by identifying the geometric properties given in the problem. The base of the pyramid is a right triangle with hypotenuse c and an acute angle 30^circ. Therefore, let ABC be our right triangle where (angle ACB = 90^circ).2. Given the right triangle with an acute angle of 30^circ, we know from trigonometry that the opposite side to the 30^circ angle (which is the side adjacent to the right angle, BC) is half the hypotenuse: [ BC = frac{c}{2} ] Similarly, the side adjacent to the 30^circ angle (which is the side opposite the right angle, AC) is: [ AC = frac{c sqrt{3}}{2} ]3. The area of the base triangle (S_{text{base}}) can be calculated using the formula for the area of a right triangle: [ S_{text{base}} = frac{1}{2} cdot AC cdot BC = frac{1}{2} cdot frac{c sqrt{3}}{2} cdot frac{c}{2} = frac{c^2 sqrt{3}}{8} ]4. We are to determine the height of the pyramid, which is given as the perpendicular distance from the apex (S) to the base (O). Given that the slant height of the pyramid (length of the side edge) is inclined at 45^circ to the base, and recognizing that (O) is the centroid of the triangle since AO = OB, we can use trigonometric properties to find (SO).5. In triangle ASO, where (angle SAO = 45^circ) and (AO = frac{c}{2}): [ SO = AO = frac{c}{2} ]6. Finally, the volume (V) of the pyramid can be calculated using the formula: [ V = frac{1}{3} cdot S_{text{base}} cdot SO = frac{1}{3} cdot frac{c^2 sqrt{3}}{8} cdot frac{c}{2} = frac{c^3 sqrt{3}}{48} ]Conclusion:[boxed{frac{c^3 sqrt{3}}{48}}]

🧠:Okay, so I need to find the length of the arc for the curve given by x = 0.25 y² - 0.5 ln y between y = 1 and y = 1.5. Hmm, arc length. I remember that the formula for arc length when the curve is expressed as x in terms of y is the integral from y = a to y = b of sqrt(1 + (dx/dy)²) dy. Let me make sure I recall that correctly. Yeah, if the curve is x = f(y), then the arc length L is ∫[a to b] sqrt(1 + (f’(y))²) dy. So, that seems right.First, I need to compute the derivative of x with respect to y. Let's see, x = 0.25 y² - 0.5 ln y. Let me compute dx/dy. The derivative of 0.25 y² with respect to y is 0.5 y. Then, the derivative of -0.5 ln y with respect to y is -0.5 * (1/y). So, putting that together, dx/dy = 0.5 y - 0.5 / y. Okay, that's straightforward.Then, I need to square that derivative. Let's compute (dx/dy)². So, (0.5 y - 0.5 / y)². Let me expand that. Let me factor out the 0.5 first: 0.5 (y - 1/y). Then, squaring it would be (0.5)² (y - 1/y)² = 0.25 (y² - 2 * y * 1/y + 1/y²) = 0.25 (y² - 2 + 1/y²). Wait, is that right? Let's check again. Expanding (y - 1/y)²: y² - 2*(y)*(1/y) + (1/y)² = y² - 2 + 1/y². Yes, that's correct. So, then multiplying by 0.25 gives 0.25 y² - 0.5 + 0.25 / y².Now, the integrand for the arc length is sqrt(1 + (dx/dy)²). Let's compute 1 + (dx/dy)². That would be 1 + [0.25 y² - 0.5 + 0.25 / y²]. Let's add those terms up. 1 - 0.5 is 0.5, so the expression becomes 0.25 y² + 0.5 + 0.25 / y². Hmm, wait, let me make sure. So:1 + (0.25 y² - 0.5 + 0.25 / y²) = 0.25 y² + (1 - 0.5) + 0.25 / y² = 0.25 y² + 0.5 + 0.25 / y². Yes, that's correct.Now, can this expression under the square root be simplified? Let me see. 0.25 y² + 0.5 + 0.25 / y². Let's write 0.25 as (1/4). So, (1/4)y² + 1/2 + (1/4)(1/y²). Hmm. Maybe this is a perfect square? Let me check. Suppose we have something like (a y + b / y)². Let's expand that: a² y² + 2 a b + b² / y². Comparing to our expression, which is (1/4)y² + 1/2 + (1/4)/y². So, matching coefficients:a² = 1/4, so a = 1/2 or -1/2.Similarly, b² = 1/4, so b = 1/2 or -1/2.Then, 2 a b should be equal to 1/2 (the middle term). Let's see:If a = 1/2 and b = 1/2, then 2ab = 2*(1/2)*(1/2) = 2*(1/4) = 1/2. Perfect! So, ( (1/2)y + (1/2)(1/y) )² = ( (y + 1/y)/2 )² = (1/4)y² + 1/2 + (1/4)/y². Exactly the expression we have. So, the integrand simplifies to sqrt( [ (0.5 y + 0.5 / y) ]² ) = |0.5 y + 0.5 / y|. Since y is between 1 and 1.5, which are positive numbers, and y is positive, so 0.5 y + 0.5 / y is definitely positive. Therefore, the square root simplifies to 0.5 y + 0.5 / y.Therefore, the arc length integral becomes ∫[1 to 1.5] (0.5 y + 0.5 / y) dy. That's much easier to integrate than dealing with the square root of a complicated expression.So, let's compute the integral. Split it into two terms:0.5 ∫ y dy + 0.5 ∫ (1/y) dy, evaluated from 1 to 1.5.First integral: 0.5 ∫ y dy = 0.5 * (0.5 y²) = 0.25 y².Second integral: 0.5 ∫ (1/y) dy = 0.5 ln |y|. Since y is positive, it's 0.5 ln y.Therefore, combining both terms, the antiderivative is 0.25 y² + 0.5 ln y. Then, evaluate from y = 1 to y = 1.5.So, compute [0.25*(1.5)² + 0.5 ln(1.5)] - [0.25*(1)² + 0.5 ln(1)].Compute each part:First, at y = 1.5:0.25*(1.5)^2 = 0.25*(2.25) = 0.56250.5 ln(1.5) ≈ 0.5 * 0.4054651 ≈ 0.20273255So, sum is 0.5625 + 0.20273255 ≈ 0.76523255At y = 1:0.25*(1)^2 = 0.250.5 ln(1) = 0.5 * 0 = 0Sum is 0.25 + 0 = 0.25Subtracting: 0.76523255 - 0.25 = 0.51523255So, approximately 0.5152. But let me check the exact value symbolically.Wait, let's compute exactly:0.25*(1.5)^2 = 0.25*(9/4) = 9/160.5 ln(1.5) = (1/2) ln(3/2)At y=1:0.25*(1)^2 = 1/40.5 ln(1) = 0So, the exact value is (9/16 + (1/2) ln(3/2)) - 1/4 = 9/16 - 1/4 + (1/2) ln(3/2)Convert 1/4 to 4/16: 9/16 - 4/16 = 5/16Therefore, exact value is 5/16 + (1/2) ln(3/2). So, to write that as a decimal, 5/16 is 0.3125, and (1/2) ln(3/2) ≈ 0.5 * 0.4054651 ≈ 0.20273255, so total ≈ 0.3125 + 0.20273255 ≈ 0.51523255, which matches the approximate value before.But the problem might expect an exact answer in terms of logarithms. So, 5/16 + (1/2) ln(3/2). Alternatively, using ln(3) - ln(2), so (1/2)(ln 3 - ln 2) + 5/16.But perhaps we can write it as 5/16 + (1/2) ln(3/2). That's a precise exact answer. Alternatively, maybe they want a decimal approximation? Let me check if the problem specifies. The original problem says to "calculate the length of the arc," but doesn't specify the form. Maybe either exact or approximate is okay, but in calculus problems like this, exact form is usually preferred if possible. However, sometimes they want a decimal. Hmm. Let me think.Alternatively, maybe 5/16 is 0.3125 and then (1/2) ln(3/2) is approximately 0.2027, so total approximately 0.5152. But maybe we need to give a more precise decimal? Let me compute ln(3/2) more accurately.ln(3/2) = ln(1.5). Let me compute that using a calculator. Let me recall that ln(1) = 0, ln(e) = 1, e ≈ 2.718. But 1.5 is less than e. Let's use Taylor series? Maybe not necessary, but since I need to approximate it, but perhaps better to use a calculator value.Alternatively, I can use natural logarithm values. Let me recall that ln(1.5) ≈ 0.4054651081. So, 0.4054651081 multiplied by 0.5 gives 0.20273255405. Then, adding 0.3125 gives 0.51523255405. So, approximately 0.5152. If we need more decimal places, maybe 0.51523. But maybe the problem expects an exact answer. Let me check again.Wait, perhaps there is a mistake here. Let me verify the steps again to ensure that I didn't make any errors.First, derivative dx/dy: 0.5 y - 0.5 / y. That's correct. Then, squaring it: (0.5 y - 0.5 / y)^2. Expanding, which gives 0.25 y² - 0.5 + 0.25 / y². Then adding 1 gives 0.25 y² + 0.5 + 0.25 / y². Then, recognizing that as a perfect square: (0.5 y + 0.5 / y)^2. Taking the square root gives 0.5 y + 0.5 / y. Then integrating that from 1 to 1.5.Integral of 0.5 y is 0.25 y². Integral of 0.5 / y is 0.5 ln y. Evaluated between 1 and 1.5, gives 0.25*(1.5)^2 + 0.5 ln(1.5) - [0.25*(1)^2 + 0.5 ln(1)]. Which simplifies to 0.25*(2.25) + 0.5 ln(1.5) - 0.25 - 0. That's 0.5625 + 0.5 ln(1.5) - 0.25 = 0.3125 + 0.5 ln(1.5). Which is 5/16 + (1/2) ln(3/2). All steps seem correct.Therefore, the exact value is 5/16 + (1/2) ln(3/2), and the approximate decimal is 0.5152. Depending on what's required, either form is acceptable, but in mathematics, exact forms are usually preferred. However, since the original function has decimal coefficients (0.25, 0.5), maybe a decimal answer is expected. But ln(3/2) is a transcendental number, so unless specified, exact form is better. Let me see if the problem says anything. The original problem says "Calculate the length of the arc...", so probably just box the exact answer or the approximate decimal. But let's see.Alternatively, maybe 5/16 is 0.3125, so 0.3125 + 0.5 ln(1.5). But ln(1.5) is approximately 0.4055, so 0.5 * 0.4055 ≈ 0.20275. Then, 0.3125 + 0.20275 ≈ 0.51525. So, approximately 0.51525. Rounded to, say, four decimal places, 0.5152. Or maybe three decimal places, 0.515. Alternatively, fractional form. Wait, 5/16 is 0.3125, so if we add that to (1/2) ln(3/2), which is approximately 0.2027, so 0.5152. If the problem expects an exact answer, 5/16 + (1/2) ln(3/2) is the exact value. Otherwise, 0.5152. Since in calculus problems like this, often both forms are acceptable, but perhaps the exact answer is preferred.Alternatively, let's see if the expression can be written in another way. For example, 5/16 + (ln 3 - ln 2)/2. But not sure if that's any simpler. Alternatively, factoring out 1/2: 5/16 + (1/2) ln(3/2). That's as simple as it gets.Therefore, the exact answer is 5/16 + (1/2) ln(3/2). Alternatively, combining the constants and the logarithm term. So, the problem probably expects that exact form. Let me check if 5/16 can be combined with the logarithmic term in some way, but I don't think so. So, that's the exact answer, and the approximate is about 0.5152.But wait, perhaps the problem expects the answer in terms of ln(3) and ln(2). Let me write that out. Since ln(3/2) is ln(3) - ln(2), so (1/2)(ln 3 - ln 2) + 5/16. Maybe that's a way to present it. But again, unless specified, both forms are correct. Alternatively, perhaps the answer can be written as (5 + 8 ln(3/2)) / 16. Let's see:5/16 + (1/2) ln(3/2) = (5 + 8*(1/2) ln(3/2)) / 16 = (5 + 4 ln(3/2)) / 16. Wait, no, that's not. Wait:If we have 5/16 + (1/2) ln(3/2). To combine over a common denominator, but ln(3/2) is not a fraction. So, perhaps better to leave it as 5/16 + (1/2) ln(3/2). Alternatively, write it as (5 + 8 ln(3/2)) / 16? Let me check:(1/2) ln(3/2) = (8/16) ln(3/2). Then, 5/16 + 8/16 ln(3/2) = [5 + 8 ln(3/2)] / 16. Yes, that works. So, another way to write the exact answer is (5 + 8 ln(3/2)) / 16. Maybe that's more compact. Let me verify:[5 + 8 ln(3/2)] / 16 = 5/16 + (8/16) ln(3/2) = 5/16 + (1/2) ln(3/2). Yes, that's correct. So, that's another way to write the exact answer. Which form is preferable? Maybe the problem expects the answer in terms of a single fraction, so (5 + 8 ln(3/2)) / 16. Alternatively, maybe leave it as is. Either way is correct.Alternatively, if we need to rationalize the coefficients, but I don't think that's necessary. So, I think either form is acceptable. Let me see if there's a standard form. But in any case, both expressions are equivalent. So, the exact answer is 5/16 + (1/2) ln(3/2) or (5 + 8 ln(3/2)) / 16, and the approximate decimal is around 0.5152.Therefore, depending on what's required, but given that the original equation has decimal coefficients, maybe a decimal answer is acceptable. However, since the problem is mathematical and precise, perhaps the exact form is better. Let me check if there are any calculation errors.Wait, when integrating 0.5 / y, we get 0.5 ln y, correct. Then evaluating from 1 to 1.5. At 1.5, 0.25*(1.5)^2 = 0.25*2.25 = 0.5625. 0.5 ln(1.5) ≈ 0.2027. Sum is 0.7652. At 1, 0.25*1 = 0.25, 0.5 ln(1) = 0. Sum is 0.25. Difference is 0.7652 - 0.25 = 0.5152. Correct. So, decimal is approximately 0.5152. Exact value is 5/16 + (1/2) ln(3/2). So, I think that's the answer.But let me check once more if the expression under the square root was correctly simplified. Original expression: 1 + (dx/dy)^2. Which we found to be (0.5 y + 0.5 / y)^2. Then taking square root gives 0.5 y + 0.5 / y. Since y is between 1 and 1.5, which are positive, so the expression inside the square root is positive, so absolute value is not necessary. So, correct.Therefore, the integral is straightforward. So, yes, the answer is indeed 5/16 + (1/2) ln(3/2) or approximately 0.5152.Therefore, the final answer is boxed{dfrac{5}{16} + dfrac{1}{2} ln dfrac{3}{2}}.Alternatively, if decimal is needed, boxed{0.5152}. But since the problem doesn't specify, and in mathematical contexts exact forms are preferred, the boxed exact expression is appropriate.

✅:To calculate the length of the arc of the curve ( x = 0.25 y^2 - 0.5 ln y ) between ( y = 1 ) and ( y = 1.5 ), we use the arc length formula for a curve defined by ( x = f(y) ):[L = int_{a}^{b} sqrt{1 + left( frac{dx}{dy} right)^2} , dy]1. Compute the derivative (frac{dx}{dy}): [ frac{dx}{dy} = 0.5y - frac{0.5}{y} ]2. Square the derivative: [ left( frac{dx}{dy} right)^2 = left( 0.5y - frac{0.5}{y} right)^2 = 0.25y^2 - 0.5 + frac{0.25}{y^2} ]3. Simplify the integrand: [ 1 + left( frac{dx}{dy} right)^2 = 0.25y^2 + 0.5 + frac{0.25}{y^2} = left( 0.5y + frac{0.5}{y} right)^2 ]4. Take the square root: [ sqrt{1 + left( frac{dx}{dy} right)^2} = 0.5y + frac{0.5}{y} ]5. Set up the integral for arc length: [ L = int_{1}^{1.5} left( 0.5y + frac{0.5}{y} right) dy ]6. Integrate term by term: [ int left( 0.5y right) dy = 0.25y^2 ] [ int left( frac{0.5}{y} right) dy = 0.5 ln |y| ]7. Evaluate the definite integral from 1 to 1.5: [ L = left[ 0.25y^2 + 0.5 ln y right]_{1}^{1.5} ] [ = left( 0.25(1.5)^2 + 0.5 ln 1.5 right) - left( 0.25(1)^2 + 0.5 ln 1 right) ] [ = left( 0.5625 + 0.5 ln 1.5 right) - left( 0.25 + 0 right) ] [ = 0.3125 + 0.5 ln 1.5 ]8. Express the exact answer: [ L = frac{5}{16} + frac{1}{2} ln frac{3}{2} ]The length of the arc is (boxed{dfrac{5}{16} + dfrac{1}{2} ln dfrac{3}{2}}).

🔑: Problem:Compute the length of the arc for the curve ( x = 0.25 y^2 - 0.5 ln y ) between the points ( y = 1 ) and ( y = 1.5 ).1. Determine the Arc Length Formula: The general formula for the arc length ( L ) of a function ( x = f(y) ) from ( y = a ) to ( y = b ) is given by: [ L = int_{a}^{b} sqrt{1 + left( frac{dx}{dy} right)^2} , dy ]2. Compute (frac{dx}{dy}): Given the function: [ x = 0.25 y^2 - 0.5 ln y ] We find the derivative (frac{dx}{dy}): [ frac{dx}{dy} = frac{d}{dy} left( 0.25 y^2 - 0.5 ln y right) = 0.5 y - frac{0.5}{y} ]3. Substitute and Simplify the Integrand: The integrand for the arc length is: [ sqrt{1 + left( frac{dx}{dy} right)^2} = sqrt{1 + left( 0.5 y - frac{0.5}{y} right)^2} ] Simplify the expression inside the square root: [ left( 0.5 y - frac{0.5}{y} right)^2 = 0.25 y^2 - 0.5 left(0.5 cdot y cdot frac{1}{y}right) + 0.25 cdot left(frac{1}{y}right)^2 = 0.25 y^2 - 0.25 + frac{0.25}{y^2} ] [ = 0.25 y^2 + frac{0.25}{y^2} - 0.25 = left(0.5 y + frac{0.5}{y}right)^2 - 1 ] Therefore, [ sqrt{1 + left( 0.5 y - frac{0.5}{y}right)^2} = sqrt{left(0.5 y + frac{0.5}{y}right)^2} ] [ = left|0.5 y + frac{0.5}{y}right| ] Since ( y ) ranges from 1 to 1.5 and both terms inside the absolute value are positive in this range: [ sqrt{1 + left( 0.5 y - frac{0.5}{y}right)^2} = 0.5 y + frac{0.5}{y} ]4. Set Up the Integral: [ L = int_{1}^{1.5} left(0.5 y + frac{0.5}{y} right) , dy ]5. Evaluate the Integral: [ int_{1}^{1.5} left(0.5 y + frac{0.5}{y}right) , dy = 0.5 int_{1}^{1.5} y , dy + 0.5 int_{1}^{1.5} frac{1}{y} , dy ] Evaluate each integral separately: [ int_{1}^{1.5} y , dy = left. frac{y^2}{2} right|_1^{1.5} = frac{(1.5)^2}{2} - frac{1^2}{2} = frac{2.25}{2} - frac{1}{2} = 1.125 - 0.5 = 0.625 ] [ int_{1}^{1.5} frac{1}{y} , dy = left. ln |y| right|_1^{1.5} = ln 1.5 - ln 1 = ln 1.5 ] Combine the results: [ L = 0.5 left( 0.625 + ln 1.5 right) = 0.3125 + 0.5 ln 1.5 ] Conclusion:[boxed{0.3125 + 0.5 ln 1.5}]

🧠:Okay, let me try to figure out this problem. So, the question is: If sets M and N have m and n elements respectively, how many possible mappings from M to N can be established? The options are A. m + n, B. m · n, C. m^n, D. n^m.Hmm, alright. Let's start by recalling what a mapping between two sets is. A mapping, or a function, from set M to set N assigns each element in M to exactly one element in N. So for every element in M, we choose some element in N to map it to. The question is then, how many different ways can we do this?Let me think of a simple example first. Suppose set M has 2 elements, say {a, b}, and set N has 3 elements, say {1, 2, 3}. How many functions are there from M to N?Well, for the element a in M, we can map it to 1, 2, or 3. Similarly, for element b in M, we can also map it to 1, 2, or 3. Since the choices are independent for each element in M, the total number of mappings should be 3 * 3 = 9. Let me list them out to check:1. a→1, b→12. a→1, b→23. a→1, b→34. a→2, b→15. a→2, b→26. a→2, b→37. a→3, b→18. a→3, b→29. a→3, b→3Yep, that's 9. So in this case, m=2, n=3, and the number of mappings is 3^2 = 9. So that would be n^m. Which is option D. But let me make sure this generalizes.If M has m elements and N has n elements, then each element in M has n choices in N. Since each element's choice is independent, the total number of mappings should be the product of the number of choices for each element. So that's n multiplied by itself m times, which is n^m.Alternatively, thinking in terms of exponents: for each of the m elements in M, there are n options, so it's n * n * ... * n (m times) = n^m.Wait, but let me check with another example. If M has 1 element and N has n elements, then the number of mappings should be n. That fits with n^1 = n. Similarly, if N has 1 element and M has m elements, then there's only 1 mapping, since every element in M must map to the single element in N. That's 1^m = 1, which also checks out.Looking at the options, D is n^m, which matches these examples. Let's cross-validate. If M is empty (m=0), how many mappings are there? Well, there's exactly one function from the empty set to any set N, which is the empty function. So if m=0, the answer should be 1. According to n^0, which is 1, that works. If N is empty (n=0), then there are no functions from M unless M is also empty. If M is non-empty and N is empty, there are zero functions, which also aligns with 0^m = 0 when m > 0. So the formula seems to handle edge cases correctly.Another way to think about it: a mapping can be represented as an ordered tuple where each position corresponds to an element of M, and each component of the tuple is the image of that element in N. Since there are m elements in M, the tuple has m components, each of which can be any of the n elements in N. The number of such tuples is n^m. For example, if M has 3 elements and N has 2, the number of mappings is 2^3 = 8, which would be tuples like (1,1,1), (1,1,2), (1,2,1), etc. Each position in the tuple is independent, so multiplication principle applies.Looking back at the answer options, D is n^m. The other options don't seem to fit. Let's see:A. m + n: This would be the size of the union if the sets were disjoint, but not related to the number of functions.B. m · n: This is the number of elements in the Cartesian product M × N, which is the set of all ordered pairs, but a function is more than just a single pair; it's a collection of pairs where each element of M appears exactly once. So the entire function is a subset of M × N with certain properties, but the count isn't m·n.C. m^n: This is the other exponent. But in our earlier example with m=2 and n=3, m^n would be 2^3=8, but we saw there were 9 functions, which is 3^2. So this is the reverse, which would be incorrect.Hence, the correct answer should be D. n^m.I can't think of any cases where this wouldn't hold, given the standard definition of a function between sets. So unless there's a different interpretation of "mappings"—like injective or surjective functions—but the question doesn't specify any restrictions, just general mappings (functions). So yes, the answer is D.

✅:The number of possible mappings (functions) from set ( M ) to set ( N ) is determined by assigning each of the ( m ) elements in ( M ) to one of the ( n ) elements in ( N ). For each element in ( M ), there are ( n ) independent choices, leading to a total of ( n times n times cdots times n ) (( m ) times), which is ( n^m ).Step-by-Step Explanation:1. Understanding Mappings: A mapping assigns every element in ( M ) to exactly one element in ( N ).2. Independent Choices: Each of the ( m ) elements in ( M ) has ( n ) options in ( N ).3. Multiplication Principle: The total number of mappings is ( underbrace{n times n times cdots times n}_{m text{ times}} = n^m ).4. Validation with Examples: - If ( M ) has 2 elements and ( N ) has 3, there are ( 3^2 = 9 ) mappings. - Edge cases (e.g., empty sets) confirm the formula's consistency.5. Eliminating Incorrect Options: - ( m+n ) and ( m cdot n ) do not account for independent assignments across all elements. - ( m^n ) reverses the exponent base and is incorrect.Answer: D. ( n^{m} )

🔑: Problem:If sets ( M ) and ( N ) contain ( m ) and ( n ) elements respectively, then the number of possible mappings from ( M ) to ( N ) is ().A. ( m+n )B. ( m cdot n )C. ( m^n )D. ( n^m )1. Let ( M = { a_1, a_2, ldots, a_m } ) and ( N = { b_1, b_2, ldots, b_n } ). 2. We need to find the number of possible functions ( f: M to N ). For each element ( a_i ) in ( M ), the function ( f(a_i) ) can take any value from the set ( N ). 3. For any ( a_i in M ), there are ( n ) choices for ( f(a_i) ) since there are ( n ) elements in ( N ).4. Since the choice of ( f(a_i) ) is independent for each ( i ), and there are ( m ) elements in ( M ), the total number of possible mappings is given by multiplying the number of choices for each element: [ n times n times cdots times n text{ (m times)} = n^m. ]5. Therefore, the number of possible mappings from ( M ) to ( N ) is ( n^m ).Conclusion:[boxed{D}]

🧠:Okay, so I need to show that for any curve C in the set S, which consists of solutions to the differential equation y' + a(x)y = b(x), the tangent line at the point where x = k passes through a point P_k that doesn't depend on which curve C we choose. Hmm, interesting. Let me think about how to approach this.First, let me recall that the differential equation given is a first-order linear ordinary differential equation. The standard form of such an equation is y' + P(x)y = Q(x), and the solution can be found using an integrating factor. Since a(x) and b(x) are never zero, we don't have to worry about division by zero issues here. Maybe I should start by writing down the general solution to this equation.The integrating factor would be e^(∫a(x) dx). Multiplying both sides of the equation by this integrating factor gives:e^(∫a(x) dx) y' + a(x) e^(∫a(x) dx) y = b(x) e^(∫a(x) dx)The left side is the derivative of [e^(∫a(x) dx) y], so integrating both sides with respect to x:e^(∫a(x) dx) y = ∫ b(x) e^(∫a(x) dx) dx + CTherefore, the general solution is:y = e^(-∫a(x) dx) [∫ b(x) e^(∫a(x) dx) dx + C]Where C is the constant of integration, which varies for different curves in the set S. So each specific solution curve in S corresponds to a different value of C.Now, the problem states that if we take any curve C in S and look at the tangent line at x = k on that curve, this tangent line passes through a point P_k that is the same for all such curves C. So regardless of the value of C, the tangent line at x = k must go through this fixed point P_k.To find P_k, I need to determine the equation of the tangent line at x = k for a general solution y(x), and then show that this line passes through a point whose coordinates don't depend on C.Let's start by finding the equation of the tangent line at x = k. For a curve y = y(x), the tangent line at x = k has the form:y = y(k) + y'(k)(x - k)So we need expressions for y(k) and y'(k). But since y satisfies the differential equation y' + a(x)y = b(x), we can express y'(k) in terms of y(k):y'(k) = b(k) - a(k) y(k)Therefore, the tangent line equation becomes:y = y(k) + [b(k) - a(k) y(k)](x - k)Now, let's denote the coordinates of a point on this tangent line as (x, y). For the tangent line to pass through a point P_k = (p, q) independent of C, substituting x = p and y = q into the equation must hold true for all values of C. That is:q = y(k) + [b(k) - a(k) y(k)](p - k)This equation must hold for any solution y(x) in S. However, different solutions y(x) have different constants C, so y(k) can take various values depending on C. Therefore, the only way this equation can hold for all y(k) is if the coefficients of y(k) cancel out, leaving an equation that doesn't depend on y(k). Let's rearrange the equation:q = y(k) [1 - a(k)(p - k)] + b(k)(p - k)To make this equation independent of y(k), the coefficient of y(k) must be zero. Therefore:1 - a(k)(p - k) = 0Which implies:p - k = 1/a(k) => p = k + 1/a(k)Then, substituting this back into the equation for q:q = 0 + b(k)(p - k) = b(k)(1/a(k)) = b(k)/a(k)Therefore, the point P_k has coordinates (k + 1/a(k), b(k)/a(k)), which do not depend on C. Hence, regardless of the solution curve C in S, the tangent line at x = k passes through the fixed point P_k = (k + 1/a(k), b(k)/a(k)).Wait, let me check this again. If I set the coefficient of y(k) to zero, then the remaining term is b(k)(p - k). But p was determined to be k + 1/a(k), so substituting that into the remaining term gives b(k)*(1/a(k)), which is indeed independent of C. Therefore, q = b(k)/a(k). Thus, the coordinates (p, q) are (k + 1/a(k), b(k)/a(k)), which only depend on k, a(k), and b(k), but not on the constant C. Since a(x) and b(x) are given functions, and k is a fixed point, P_k is indeed independent of the curve C. That makes sense.To verify, let's consider a specific example. Suppose a(x) is a constant function, say a(x) = a, and b(x) is another constant function, b(x) = b. Then the differential equation is y' + a y = b. The general solution would be y = (b/a) + C e^(-a x). Taking the derivative y' = -a C e^(-a x). At x = k, y(k) = (b/a) + C e^(-a k), and y'(k) = -a C e^(-a k). Then the tangent line at x = k is:y = [ (b/a) + C e^(-a k) ] + [ -a C e^(-a k) ](x - k)Simplifying:y = (b/a) + C e^(-a k) [1 - a(x - k)]If we set 1 - a(x - k) = 0, then x = k + 1/a. Plugging this x into the equation for y:y = (b/a) + C e^(-a k) * 0 = (b/a)Therefore, the point (k + 1/a, b/a) is on every tangent line at x = k, regardless of C. This matches the general result we found earlier, since in this case a(k) = a and b(k) = b, so P_k = (k + 1/a, b/a). Therefore, the example confirms the conclusion.Another check: suppose a(x) and b(x) are functions. Let’s take a(x) = x and b(x) = x^2. Then, according to the formula, P_k would be (k + 1/k, x^2 / x evaluated at x = k) = (k + 1/k, k). Wait, b(k)/a(k) = k^2 / k = k. So P_k is (k + 1/k, k). Let's see if that works.The differential equation is y' + x y = x^2. The integrating factor is e^(∫x dx) = e^(x²/2). Multiplying through:e^(x²/2) y' + x e^(x²/2) y = x^2 e^(x²/2)Integrate both sides:e^(x²/2) y = ∫ x^2 e^(x²/2) dx + CHmm, integrating x^2 e^(x²/2) dx might be tricky. Let me see. Let’s make substitution u = x, dv = x e^(x²/2) dx. Wait, maybe integration by parts. Let’s set u = x, dv = x e^(x²/2) dx. Then du = dx, and v = e^(x²/2). So:∫ x^2 e^(x²/2) dx = x e^(x²/2) - ∫ e^(x²/2) dxBut ∫ e^(x²/2) dx is not elementary. Hmm, so the integral can’t be expressed in terms of elementary functions. However, the general solution would be:y = e^(-x²/2) [x e^(x²/2) - ∫ e^(x²/2) dx + C] = x - e^(-x²/2) ∫ e^(x²/2) dx + C e^(-x²/2)But even without computing the exact solution, let's check the tangent line at x = k. For any solution y(x), the tangent line at x = k is:y = y(k) + y'(k)(x - k)But from the differential equation, y'(k) = b(k) - a(k) y(k) = k^2 - k y(k). So the tangent line equation is:y = y(k) + (k^2 - k y(k))(x - k)Simplify:y = y(k) [1 - k(x - k)] + k^2 (x - k)Now, to find the fixed point P_k, we need this equation to hold for all y(k). Let's rearrange:y = y(k) [1 - k x + k^2] + k^2 x - k^3Wait, actually, expanding the terms:y = y(k) + k^2 (x - k) - k y(k)(x - k)Grouping terms with y(k):y = y(k) [1 - k(x - k)] + k^2 (x - k)To have this pass through a fixed point regardless of y(k), the coefficient of y(k) must be zero when evaluated at P_k. So setting 1 - k(x - k) = 0 gives x = k + 1/k. Then substituting x = k + 1/k into the equation:y = 0 + k^2 (k + 1/k - k) = k^2 (1/k) = kTherefore, the fixed point P_k is (k + 1/k, k), which matches our earlier formula since a(k) = k and b(k) = k^2, so P_k = (k + 1/k, k^2/k) = (k + 1/k, k). Thus, even in this case, the tangent lines at x = k all pass through (k + 1/k, k), independent of C. That checks out.Therefore, the reasoning seems solid. The key idea is that the tangent line equation at x = k can be written in terms of y(k) and y'(k), which are related by the differential equation. By expressing y'(k) in terms of y(k), the tangent line equation becomes a linear equation in y(k). For this equation to hold for all values of y(k) (i.e., for all solutions C), the coefficients of y(k) must vanish, leading to a fixed point that doesn't depend on C. This point is determined solely by the functions a(x) and b(x) evaluated at x = k, and the value of k itself.Another way to look at it is geometrically. The set S consists of all solutions to the linear differential equation, which are a family of curves differing by the constant C. The condition that their tangent lines at x = k pass through a common point suggests that this point is a sort of "focus" or "pivot" point for all these curves at that particular x-value. This is a property inherent to the structure of the differential equation itself, not individual solutions.To further ensure that no mistakes were made, let's consider if there's any step that might have an error. For example, when solving for q, after setting the coefficient of y(k) to zero, we found q = b(k)(p - k). Then, since p was found as k + 1/a(k), substituting that gives q = b(k)/a(k). This substitution is correct because p - k = 1/a(k). Therefore, the coordinates (p, q) are correctly calculated.Additionally, considering the example where a(x) and b(x) are constants confirmed the result, which gives more confidence. The second example with a(x) = x and b(x) = x^2 also worked out, showing that the result holds even when a(x) and b(x) are not constants.One potential concern might be if a(k) is zero, but the problem statement specifies that a(x) is never zero, so we don't have to worry about division by zero. Similarly, b(x) is never zero, though in the final expression for P_k, q = b(k)/a(k) could be non-zero even if b(k) were allowed to be zero, but since b(k) is non-zero here, q is also non-zero. However, the problem doesn't state that P_k is non-zero, just that it's independent of C, which holds.In summary, the steps are:1. Recall the general solution of the linear ODE, which includes a constant C.2. For a specific solution y(x), find the tangent line at x = k, which involves y(k) and y'(k).3. Use the ODE to express y'(k) in terms of y(k).4. Substitute into the tangent line equation and rearrange terms.5. Determine the conditions (setting coefficients of y(k) to zero) required for the tangent line to pass through a fixed point regardless of C.6. Solve these conditions to find P_k = (k + 1/a(k), b(k)/a(k)).7. Verify with examples to confirm the result.Therefore, the conclusion is proven.

✅:To demonstrate that the tangent at ( x = k ) on any curve ( C in S ) passes through a fixed point ( P_k ) independent of ( C ), follow these steps:1. General Solution of the ODE: The differential equation ( y' + a(x)y = b(x) ) has the general solution: [ y = e^{-int a(x) , dx} left( int b(x) e^{int a(x) , dx} , dx + C right), ] where ( C ) is the constant of integration.2. Tangent Line Equation: For a solution ( y(x) ), the tangent line at ( x = k ) is: [ y = y(k) + y'(k)(x - k). ] Using the ODE ( y'(k) = b(k) - a(k)y(k) ), substitute ( y'(k) ): [ y = y(k) + left[ b(k) - a(k)y(k) right](x - k). ]3. Eliminating ( C )-Dependence: Rearrange the tangent line equation: [ y = y(k)left[1 - a(k)(x - k)right] + b(k)(x - k). ] For this line to pass through a fixed point ( P_k = (p, q) ), the coefficient of ( y(k) ) must vanish to eliminate ( C )-dependence: [ 1 - a(k)(p - k) = 0 implies p = k + frac{1}{a(k)}. ] Substitute ( p ) into the remaining term for ( q ): [ q = b(k)left(p - kright) = frac{b(k)}{a(k)}. ]4. Conclusion: The coordinates of ( P_k ) are: [ P_k = left( k + frac{1}{a(k)}, frac{b(k)}{a(k)} right). ] These depend only on ( a(k) ), ( b(k) ), and ( k ), not on ( C ).Final Answer:The tangent at ( x = k ) on any curve ( C in S ) passes through the point ( boxed{left( k + frac{1}{a(k)}, frac{b(k)}{a(k)} right)} ), which is independent of ( C ).

🔑:To demonstrate that the tangent at the point ( x = k ) on a curve ( C in S ) passes through a point ( P_k ) which is independent of ( C ), we follow these steps:1. Definition of the curve (C): Consider the differential equation defining the curve: [ y' + a(x) y = b(x) ] where ( a(x) ) and ( b(x) ) are never zero.2. Equation of the tangent line at ( (k, y(k)) ): Let ( C_h ) be the specific curve within the set ( S ) such that ( y(k) = h ). We want to find the equation of the tangent line to this curve at the point ((k, h)).3. Gradient at ((k, h)): From the differential equation, the slope (gradient) of the tangent line at ( x = k ) is given by the derivative evaluated at ( x = k ): [ y'(k) = b(k) - a(k) y(k) ] Since ( y(k) = h ), we substitute to find: [ y'(k) = b(k) - a(k) h ]4. Equation of the tangent line: Using the point-slope form of the line equation, the tangent at ((k, h)) is: [ y - h = (b(k) - a(k) h)(x - k) ] Rearranging and simplifying, we get: [ y - h = b(k)x - b(k)k - a(k)hx + a(k)hk ] [ y - h = b(k)x - b(k)k - a(k)hx + a(k)hk ] Isolating (h), we get: [ a(k)x h - a(k) k h = b(k)x - b(k)k - y + h ] [ (a(k)x - a(k)k) h = b(k)(x - k) - y + h ]5. Identifying intersection point ( P_k ): For the tangent to always pass through a fixed point ((P_x, P_y)), independent of (h), we should identify this point. By examining the form: [ (a(k)x - a(k)k - 1) h = (b(k)(x - k) - y) ] We solve for ( x ) and ( y ) to find that the tangent must pass through the point: [ x = k + frac{1}{a(k)},quad y = frac{b(k)}{a(k)} ]6. Conclusion: Hence, the tangent at ( x = k ) on any curve ( C in S ) always passes through the point ( left( k + frac{1}{a(k)}, frac{b(k)}{a(k)} right) ), which is indeed independent of the specific curve: [ boxed{left( k + frac{1}{a(k)}, frac{b(k)}{a(k)} right)} ]

🧠:Okay, let's try to figure out this problem step by step. So, the competition has 25 questions, right? For each correct answer, you get 4 points, and for each incorrect or unanswered question, you lose 1 point. The goal is to find the minimum number of correct answers needed to score at least 60 points. Hmm, let me think.First, let me make sure I understand the scoring system. If a student answers a question correctly, they gain 4 points. If they answer incorrectly or leave it blank, they lose 1 point. So, even if they don't answer, they still get a penalty. That means every question that isn't correct is costing them a point. Alright, so the total number of questions is 25. Let's denote the number of correct answers as 'x'. Then, the number of incorrect or unanswered questions would be 25 - x. Because each correct gives +4 and each incorrect gives -1, the total score can be calculated as:Total score = (Number of correct answers × 4) + (Number of incorrect answers × -1)So, substituting the variables:Total score = 4x - (25 - x)Simplify that:4x - 25 + x = 5x - 25So the total score is 5x - 25. We need this total score to be at least 60. So:5x - 25 ≥ 60Let me solve that inequality. First, add 25 to both sides:5x ≥ 85Then, divide both sides by 5:x ≥ 17Wait, so x needs to be at least 17? Let me check that again because 17 × 4 = 68, and then subtract the incorrect ones. If x is 17, then incorrect is 8. So, 17*4 = 68, minus 8*1 = 8, so 68 - 8 = 60. Exactly 60. So, if you get 17 correct, you reach exactly 60. So that's the minimum?But hold on, let me verify this with an example. Suppose a student answers 17 correctly and 8 incorrectly. Then, the calculation is 17*4 = 68, and 8*-1 = -8, so total is 60. That works. If they answer 16 correctly, then 16*4 = 64, and 9*-1 = -9, so total is 64 -9 = 55. Which is below 60. So 16 correct answers give 55 points, which is insufficient. Therefore, 17 is indeed the minimum.But wait, let me make sure that there isn't a scenario where maybe skipping some questions would result in a higher score. For example, if you answer 17 correctly and leave 8 blank, you get 60. But what if you answer some wrong and some correct? Wait, no, because each incorrect or unanswered question deducts 1 point. So, whether you answer incorrectly or leave it blank, the effect is the same. So, the total deduction is just the number of questions not answered correctly.Therefore, to maximize the score, you want as many correct answers as possible, and as few incorrect/unanswered as possible. But here we need the minimal number of correct answers. So, we set up the equation where the total score is 60, and solve for x. Since 17 gives exactly 60, and 16 gives 55, which is less, 17 is indeed the minimum.Wait, but let's think again. Maybe there's a way to get 60 points with fewer correct answers if some questions are left blank? Wait, no. Because leaving a question blank is the same as answering incorrectly in terms of scoring. Both deduct 1 point. So, whether you answer 17 correctly and 8 incorrectly, or 17 correctly, 0 incorrect, and 8 blank, the result is the same: 17*4 -8*1 = 60. So, the number of correct answers is the key variable here. The rest, whether incorrect or blank, don't matter beyond their quantity.Therefore, the minimal number of correct answers needed is 17. Hmm. That seems straightforward, but let me check another angle. Suppose a student answers some questions correctly and some incorrectly, and skips some. Wait, but again, incorrect and skipped are treated the same. So, suppose a student answers x correct, y incorrect, and z unanswered. Then x + y + z =25, and total score is 4x - y - z. But since y + z =25 - x, the total score is 4x - (25 - x) =5x -25. Therefore, regardless of how they split between y and z, the total score only depends on x. Therefore, the minimal x is 17.So, yeah, the answer is 17. But let me test with 17.5, even though you can't answer half a question. Wait, 5x -25 ≥60. Solving for x gives x ≥17. Since x must be an integer, 17 is the minimal. So that's confirmed.But maybe there's a trick here? Like, is there a different way to interpret the problem? For example, does "incorrect answer" mean attempted but wrong, and "unanswered" is separate? But according to the problem statement, it says "each incorrect answer (including unanswered questions) deducts 1 point." Wait, so actually, the problem says that both incorrect answers and unanswered questions deduct 1 point. So, in other words, whether you answer incorrectly or don't answer at all, you lose 1 point. So, in that case, the number of incorrect answers plus the number of unanswered questions is 25 - x, where x is the number of correct answers. Therefore, the total deduction is (25 - x)*1, which is subtracted from the total correct points (4x). So, the formula is indeed 4x - (25 - x) =5x -25. So, as before, 5x -25 ≥60 leads to x≥17. So, 17.Therefore, the minimal number of correct answers is 17. I think that's solid. Let me just check with 17:17 correct, 8 incorrect/unanswered: 17*4=68, 8*-1=-8, total 60.If you have 18 correct, 7 incorrect: 18*4=72, 7*-1=-7, total 65. Which is above 60, but 17 is the minimum. So, 17 is the answer.But hold on, wait, suppose a student answers 17 correct, but then tries some questions and gets them wrong. Does that affect anything? Wait, no, because even if you answer some wrong, you still get the same deduction as leaving them blank. So, if you answer 17 correct, 3 wrong, and 5 blank, then total deduction is 3 +5=8, same as 8 wrong or 8 blank. So, regardless, the deduction is 8. So, total is 68 -8=60. So, same result.Therefore, regardless of how the student distributes their incorrect answers and blanks, the total deduction is 25 - x. Therefore, the total score is 5x -25, which must be ≥60. Therefore, x must be ≥17. So, the answer is 17.I think that's thorough. I considered different scenarios, checked the formula, tested with numbers, considered edge cases, but all lead to 17. Therefore, the minimum number of correct answers needed is 17.Final AnswerThe minimum number of questions a student needs to answer correctly is boxed{17}.

✅:To determine the minimum number of correct answers needed to qualify for the second test, we start by defining the number of correct answers as ( x ). Each correct answer earns 4 points, and each incorrect or unanswered question deducts 1 point. The total number of questions is 25, so the number of incorrect or unanswered questions is ( 25 - x ).The total score can be calculated using the formula:[text{Total score} = 4x - (25 - x)]Simplifying this expression, we get:[text{Total score} = 4x - 25 + x = 5x - 25]We need this score to be at least 60 points:[5x - 25 geq 60]Solving for ( x ):[5x geq 85 implies x geq 17]We verify this by checking the score for ( x = 17 ):- 17 correct answers: ( 17 times 4 = 68 ) points- 8 incorrect or unanswered questions: ( 8 times (-1) = -8 ) points- Total score: ( 68 - 8 = 60 ) pointsIf ( x = 16 ), the score would be:- 16 correct answers: ( 16 times 4 = 64 ) points- 9 incorrect or unanswered questions: ( 9 times (-1) = -9 ) points- Total score: ( 64 - 9 = 55 ) points (insufficient)Thus, the minimum number of correct answers needed to score at least 60 points is (boxed{17}).

🔑:To determine the minimum number of correct answers required for a student to score at least 60 points, we follow these steps:1. Calculate the total maximum score if all answers are correct: [ 4 text{ points per correct answer} times 25 text{ questions} = 100 text{ points} ]2. Define the scoring and penalty system: - Each correct answer scores 4 points. - Each incorrect or unanswered question results in a deduction of 1 point.3. Calculate the total deduction for an incorrect answer: - For each incorrect answer, we lose 4 points for not scoring and an additional 1 point penalty. - The total deduction per incorrect answer is: [ 4 text{ points scored if correct} + 1 text{ points penalty} = 5 text{ points lost} ]4. Determine the maximum allowable total deduction to still pass: - The minimum passing score is 60 points. - The maximum allowable decrease from the full score (100 points) to get exactly 60 points is: [ 100 text{ points} - 60 text{ points} = 40 text{ points} ]5. Find the maximum number of incorrect answers allowed: - Each incorrect answer results in a deduction of 5 points. - Thus, the maximum number of incorrect answers that would allow passing is calculated as: [ frac{40 text{ points}}{5 text{ points per incorrect answer}} = 8 text{ incorrect answers} ]6. Calculate the minimum number of correct answers required: - If there can be at most 8 incorrect answers out of 25 questions, the minimum number of correct answers must be: [ 25 text{ total questions} - 8 text{ incorrect answers} = 17 text{ correct answers} ] Conclusion:[boxed{17}]

🧠:Okay, so I need to find the remaining volume of an 8 cm cube after a 4 cm square hole has been cut through its center. Hmm, let me think. The cube's volume is straightforward—side length cubed. So original volume is 8 cm * 8 cm * 8 cm = 512 cm³. But then there's this square hole that's been removed. The problem is figuring out the volume of the hole to subtract from the original cube.Wait, the hole is a square hole with 4 cm sides. But how is it oriented? Since it's cut through the center, I assume it goes through the entire cube. So, if it's a square hole, maybe it's a rectangular prism? But then, if the cube is 8 cm tall, and the hole is 4 cm square, does that mean the hole is 4 cm in two dimensions and 8 cm in the third? But that would be a rectangular prism with dimensions 4 cm x 4 cm x 8 cm. So the volume of the hole would be 4*4*8 = 128 cm³. Then subtracting that from 512 gives 512 - 128 = 384 cm³. Hmm, but 384 is one of the options, option D. But wait, maybe that's not correct.Wait, hold on. Maybe the hole isn't a straight rectangular prism. If it's a square hole cut through the center, perhaps it's more complicated because the hole might intersect the cube in such a way that the removal isn't just a simple prism. Let me visualize the cube. An 8 cm cube. If you cut a square hole through the center, does the hole go through all six faces? Or just two opposite faces?Usually, when they say a hole is cut through the center, it's through two opposite faces. So, if the hole is a square of 4 cm, then the cross-section is 4 cm x 4 cm, and the length of the hole is equal to the length of the cube, which is 8 cm. So that would make the hole's volume 4*4*8 = 128, as before. So 512 - 128 = 384. So D? But I feel like maybe that's too straightforward.Wait, but maybe the hole is not aligned with the faces. If the hole is drilled through the center but along a diagonal or something, then the calculation would be different. But the problem says it's a square hole. So if it's a square hole, the cross-section must be aligned such that the square's sides are parallel to the cube's edges. Otherwise, the cross-section wouldn't be a square. Because if you drill a square hole along a diagonal, the cross-section might not remain a square.Therefore, the hole is probably a straight square prism (rectangular prism with square cross-section) going through the cube from one face to the opposite face, aligned with the cube's edges. Therefore, the volume removed is 4*4*8 = 128. Then remaining volume is 512 - 128 = 384. So answer D. But wait, let me check again.Alternatively, maybe the hole is through the center in all three dimensions? Like a 3D cross? So if you have a cube, and you remove a square hole along each axis? But that would be more complicated. But the problem states a 4 cm square hole cut through its center. So probably a single hole, not three orthogonal holes.But just to be thorough, let me think. If it's a single square hole through the center, going through two opposite faces. The cross-section is 4 cm x 4 cm, length 8 cm. So volume 128. Remaining volume 384. But the answer choices include 384 as D, but also 256 as C. Maybe I need to consider something else.Wait, perhaps the hole is not 8 cm long? If the hole is a square prism that goes through the cube, but the cube is 8 cm in each dimension. If the hole is 4 cm in cross-section, maybe the length isn't 8 cm. Wait, no. If you have a cube and you drill a hole through its center from one face to the opposite face, the length of the hole is equal to the side length of the cube. So 8 cm. Therefore, the hole's volume is 4*4*8=128. Therefore, 512-128=384. So answer D.But maybe the hole is in the center but doesn't go through the entire cube? Wait, the problem says "a 4 cm square hole cut through its center". The phrase "cut through" suggests that it goes all the way through. So length is 8 cm.Alternatively, maybe the hole is a cube of 4 cm removed from the center? But then the volume would be 4*4*4=64. Then remaining volume 512-64=448, which is E. But the problem says a square hole, not a cube hole. A square hole is typically a prism with square cross-section, not a cube. So if it's a square hole, the cross-section is 4x4, and the length is 8. So volume 128.Alternatively, maybe the hole is 4 cm in each dimension, but since it's in the center, perhaps it's only 4 cm long? Wait, but if the cube is 8 cm, and you have a hole through the center, you have to go from one side to the other. So if the hole is 4 cm in cross-section, but centered, then the length is 8 cm. For example, imagine a cube of 8 cm. If you have a square hole in the middle, with sides 4 cm, aligned with the cube's edges. So the hole starts at the center and extends 2 cm in each direction? Wait, no. Wait, the hole is 4 cm square. So the hole's cross-section is 4 cm x 4 cm. To have a 4 cm square hole through the center, the hole must pass through the entire cube. So if the cube is 8 cm long, then the hole is 8 cm long. So 4x4x8=128.Alternatively, if the hole is only in the center, not passing through the entire cube. But the problem says "cut through its center", which probably means passing through the cube. So the hole is passing through the center, going from one face to the opposite face. So the length is 8 cm, cross-section 4x4. So volume 128. Thus remaining volume 384. So answer D.Wait, but let me check again. Maybe I'm missing something. Suppose the hole is a square prism of 4x4x8, passing through the cube. So when you remove that, you subtract 128 from 512, getting 384. But maybe when you remove the hole, you are also removing some overlapping parts? Wait, no, because the hole is entirely within the cube. Since the cube is 8 cm, and the hole is 4 cm in cross-section, centered. So the hole is positioned such that its sides are 2 cm away from each face (since 8 - 4 = 4, divided by 2 is 2 cm on each side). So the hole is 4x4 in cross-section, 8 cm long. So yes, 128 cm³. Then remaining volume is 512 - 128 = 384. So answer D. Therefore, the answer is D) 384.But wait, wait. Let me visualize this again. The cube is 8 cm on each side. The square hole is 4 cm on each side. If the hole is drilled through the center, along one axis, say the vertical axis, then looking at the cube from the top, the hole is a 4x4 square in the center. So from the top view, the hole is 4x4, and it goes all the way through the 8 cm height. Therefore, the volume is indeed 4x4x8=128. Subtracted from 512 gives 384. So D.Alternatively, maybe the hole is in the center but in three dimensions? For example, like a 3D cross where three orthogonal square holes intersect at the center. But the problem states "a 4 cm square hole", singular. So it's one hole, not three. So that's not the case. So yes, D seems correct.But let me check the answer options again. The options are (A) 64, (B) 128, (C) 256, (D) 384, (E) 448. Our calculation leads to D) 384, which is an option. So I think that's the correct answer. But maybe the problem is trickier. Let me see if there's another way.Wait, maybe the hole is a square hole in 3D, like a cube. But the problem says a square hole. A square hole is two-dimensional, so extruded through the third dimension. So a square prism. So yeah, 4x4x8. So 128. So 512-128=384. So D.Alternatively, maybe the hole is a square tunnel that goes through the cube not along one axis but through the center in another way? Like, through the middle of edges or something. But in that case, the cross-section might not be a square. Hmm.Wait, if you drill a square hole through the center of the cube but along the space diagonal, then the cross-section would not be a square. So the problem states it's a square hole, so the cross-section must be a square. Therefore, the hole must be aligned with the cube's faces.Therefore, the hole is a straight square prism, aligned with one axis, with cross-section 4x4 cm², and length 8 cm. Therefore, the volume is 4*4*8=128. Therefore, remaining volume is 512-128=384. So D.Alternatively, maybe the 4 cm square hole refers to the diagonal of the square? Wait, no. A square hole with 4 cm square—should mean each side is 4 cm. If it were the diagonal, it would say 4 cm diagonal.Alternatively, maybe the hole is such that the square is 4 cm in perimeter. But 4 cm perimeter would be 1 cm per side, which seems too small. So that's not the case.Alternatively, maybe the cross-sectional area is 4 cm². Then the volume would be 4*8=32. But the problem says a 4 cm square hole. A 4 cm square would mean each side is 4 cm. So cross-section area is 16 cm². Then volume is 16*8=128. Yeah, so that's the same as before.Therefore, the answer is D) 384. So I think that's correct. Wait, but let me check once more.Original volume: 8^3=512.Hole volume: 4x4x8=128.Remaining: 512-128=384. Yes. So D.But the answer is given as an option, and D is 384. So that's the answer.Final Answerboxed{D}

✅:The original volume of the (8 mathrm{~cm}) cube is calculated as follows:[8 mathrm{~cm} times 8 mathrm{~cm} times 8 mathrm{~cm} = 512 mathrm{~cm}^3]A (4 mathrm{~cm}) square hole is cut through the center of the cube. This hole is a rectangular prism with a square cross-section of side (4 mathrm{~cm}) and a length equal to the side length of the cube, which is (8 mathrm{~cm}). The volume of this hole is:[4 mathrm{~cm} times 4 mathrm{~cm} times 8 mathrm{~cm} = 128 mathrm{~cm}^3]Subtracting the volume of the hole from the original volume of the cube gives the remaining volume:[512 mathrm{~cm}^3 - 128 mathrm{~cm}^3 = 384 mathrm{~cm}^3]Thus, the remaining volume of the cube is (boxed{D}).

🔑:Given that an (8 text{ cm}) cube has a (4 text{ cm}) square hole cut through its center, we need to determine the remaining volume in (text{cm}^3).1. Calculate the volume of the original cube: The side length of the cube is (8 text{ cm}). The volume of a cube is given by the formula (V = a^3), where (a) is the side length. [ V_{text{cube}} = 8^3 = 512 text{ cm}^3 ]2. Calculate the volume of the hole through the center: The hole is a (4 text{ cm}) square hole that goes through the entire (8 text{ cm}) length of the cube. The cross-sectional area of the hole is (4 text{ cm} times 4 text{ cm} = 16 text{ cm}^2). The volume of the hole is then the cross-sectional area multiplied by the length through the cube. [ V_{text{hole}} = A_{text{cross-section}} times text{length} = 16 text{ cm}^2 times 8 text{ cm} = 128 text{ cm}^3 ]3. Subtract the volume of the hole from the volume of the original cube: The remaining volume of the cube after the hole is cut through it is the volume of the original cube minus the volume of the hole. [ V_{text{remaining}} = V_{text{cube}} - V_{text{hole}} = 512 text{ cm}^3 - 128 text{ cm}^3 = 384 text{ cm}^3 ] Conclusion:The remaining volume of the cube, with the hole through its center, is:[boxed{384}]