💡Okay, so I have this problem here: the function is f(x) = log base a of x plus x minus b, where a is greater than 0 and not equal to 1. The conditions given are that 2 < a < 3 and 3 < b < 4. I need to find the interval (n, n+1) where the zero of the function, x₀, lies, and n is a positive integer. The options are A:1, B:2, C:3, D:4.Alright, let me break this down. First, I know that to find the zero of a function, I need to find the value of x where f(x) = 0. So, I need to solve log_a(x) + x - b = 0. That means log_a(x) + x = b.Given that a is between 2 and 3, and b is between 3 and 4. Hmm, okay. So, maybe I can test some values of x to see where the function crosses zero.Let me think about the behavior of the function. The function f(x) = log_a(x) + x - b is a combination of a logarithmic function and a linear function. The logarithmic function, log_a(x), is increasing because a > 1. The linear function, x - b, is also increasing. So, overall, f(x) is an increasing function because both components are increasing.Since f(x) is increasing, it can have at most one zero. So, there will be exactly one x where f(x) = 0.Now, I need to find the interval where this zero lies. Let's try plugging in some integer values for x and see the sign of f(x). That should help me figure out where the zero is.Let's start with x = 2.f(2) = log_a(2) + 2 - b.I know that a is between 2 and 3. So, log_a(2) is less than 1 because log base a of a is 1, and 2 is less than a. So, log_a(2) < 1.Therefore, f(2) < 1 + 2 - b = 3 - b.But b is between 3 and 4, so 3 - b is negative. Therefore, f(2) is less than a negative number, which means f(2) is negative.Okay, so f(2) < 0.Now, let's try x = 3.f(3) = log_a(3) + 3 - b.Again, a is between 2 and 3. So, log_a(3) is greater than 1 because 3 is greater than a. Specifically, since a is less than 3, log_a(3) is greater than 1.So, log_a(3) > 1.Therefore, f(3) > 1 + 3 - b = 4 - b.But b is between 3 and 4, so 4 - b is between 0 and 1. Therefore, f(3) is greater than a positive number, which means f(3) is positive.So, f(3) > 0.Since f(x) is continuous (both log_a(x) and x - b are continuous for x > 0), and it's increasing, by the Intermediate Value Theorem, there must be a zero between x = 2 and x = 3.Therefore, the zero x₀ is in the interval (2, 3). So, n is 2.Wait, let me just double-check. Maybe I should test x = 1 and x = 4 to see if the function could cross zero elsewhere.For x = 1:f(1) = log_a(1) + 1 - b = 0 + 1 - b = 1 - b.Since b > 3, 1 - b is negative. So, f(1) is negative.For x = 4:f(4) = log_a(4) + 4 - b.log_a(4) is greater than log_a(3), which we already established is greater than 1. So, log_a(4) is greater than 1, maybe around 2 if a is 2, but a is between 2 and 3, so log_a(4) is less than 2 because log base 2 of 4 is 2, and since a is greater than 2, log_a(4) is less than 2.So, log_a(4) is between 1 and 2.Therefore, f(4) is between 1 + 4 - b and 2 + 4 - b, which is between 5 - b and 6 - b.Since b is between 3 and 4, 5 - b is between 1 and 2, and 6 - b is between 2 and 3. So, f(4) is positive.So, f(4) is positive, but we already saw that f(3) is positive as well. So, the function is increasing, so it crosses zero somewhere between 2 and 3.Therefore, x₀ is in (2, 3), so n is 2.I think that's solid. So, the answer should be B:2.**Final Answer**boxed{B}

💡Alright, so I have this problem about a rectangle EFGH with an area of 2450. There's also an ellipse that passes through points E and G, and it has its foci at F and H. The area of the ellipse is given as 2450π. I need to find the perimeter of the rectangle EFGH.Hmm, okay. Let me start by recalling some properties of rectangles and ellipses. A rectangle has opposite sides equal and all angles are right angles. The area of a rectangle is length times width. So, if I let the sides of the rectangle be x and y, then the area is xy = 2450.Now, the ellipse part. The area of an ellipse is πab, where a is the semi-major axis and b is the semi-minor axis. Given that the area is 2450π, that means πab = 2450π, so ab = 2450.The ellipse passes through points E and G, which are opposite corners of the rectangle. So, E and G are the endpoints of the diagonal of the rectangle. The foci of the ellipse are at F and H, which are the other two opposite corners of the rectangle. So, the foci are located at the other two corners.I remember that in an ellipse, the distance between the foci is 2c, where c is the distance from the center to each focus, and c^2 = a^2 - b^2. So, if I can find the distance between F and H, that should be equal to 2c.But F and H are also opposite corners of the rectangle, so the distance between them is the same as the distance between E and G, which is the diagonal of the rectangle. The length of the diagonal of a rectangle can be found using the Pythagorean theorem: √(x^2 + y^2). So, the distance between F and H is √(x^2 + y^2), which is equal to 2c.So, 2c = √(x^2 + y^2). Therefore, c = √(x^2 + y^2)/2.But I also know that c^2 = a^2 - b^2. So, (√(x^2 + y^2)/2)^2 = a^2 - b^2. That simplifies to (x^2 + y^2)/4 = a^2 - b^2.Okay, so I have that equation. Now, I also know that the ellipse passes through points E and G. Since E and G are on the ellipse, the sum of the distances from E to each focus (F and H) should be equal to 2a, which is the major axis length.So, let's think about that. The distance from E to F and from E to H. Since E is a corner of the rectangle, the distance from E to F is one side of the rectangle, say x, and the distance from E to H is the diagonal, which is √(x^2 + y^2). Wait, no, that's not right. If E is one corner, F is adjacent to E, so the distance from E to F is x, and the distance from E to H would be the diagonal, which is √(x^2 + y^2). But wait, in the ellipse, the sum of the distances from any point on the ellipse to the two foci is 2a. So, for point E, which is on the ellipse, the sum of distances to F and H is 2a.So, distance from E to F is x, and distance from E to H is √(x^2 + y^2). Therefore, x + √(x^2 + y^2) = 2a.Similarly, for point G, which is the opposite corner, the distance from G to F is the diagonal √(x^2 + y^2), and the distance from G to H is y. So, √(x^2 + y^2) + y = 2a.Wait, but both E and G are on the ellipse, so both these sums should equal 2a. So, from E: x + √(x^2 + y^2) = 2a, and from G: √(x^2 + y^2) + y = 2a. Therefore, both x + √(x^2 + y^2) and y + √(x^2 + y^2) equal 2a. That suggests that x = y? Because otherwise, how can both sums be equal? Unless x = y, which would make the rectangle a square. But the area is 2450, so if it's a square, then x = y = √2450. But let me check that.Wait, no, that might not necessarily be the case. Maybe x ≠ y, but the sums still equal 2a. Let me think. If x + d = 2a and y + d = 2a, where d is the diagonal, then x + d = y + d, which implies x = y. So, that would mean the rectangle is a square. Hmm, interesting.So, if x = y, then the rectangle is a square with sides x, and area x^2 = 2450, so x = √2450. Let me compute √2450. 2450 is 49 * 50, so √2450 = 7√50 = 7*5√2 = 35√2. So, each side is 35√2, and the perimeter would be 4*35√2 = 140√2. But wait, that seems too straightforward, and the problem mentions an ellipse, so maybe it's not a square.Wait, maybe I made a wrong assumption. Let me go back. I said that from point E, the sum of distances to F and H is 2a, which is x + √(x^2 + y^2). Similarly, from point G, it's y + √(x^2 + y^2). So, both equal 2a, so x + d = y + d, which implies x = y. So, unless the rectangle is a square, this can't hold. But the problem doesn't specify that it's a square, just a rectangle. So, maybe my initial assumption is wrong.Alternatively, perhaps the ellipse is such that the major axis is along the diagonal of the rectangle. So, maybe the major axis length is equal to the diagonal of the rectangle. Let me think.Wait, the ellipse passes through E and G, which are endpoints of the diagonal. So, the major axis might be the diagonal. So, the major axis length is 2a, which would be equal to the diagonal of the rectangle, which is √(x^2 + y^2). So, 2a = √(x^2 + y^2). Therefore, a = √(x^2 + y^2)/2.But earlier, I had that 2c = √(x^2 + y^2), so c = √(x^2 + y^2)/2, which is equal to a. Wait, that can't be, because in an ellipse, c < a. So, that suggests that my assumption that 2a is the diagonal is incorrect.Wait, maybe the major axis is not the diagonal. Let me think again. The ellipse passes through E and G, which are opposite corners, so the major axis might be the line connecting E and G, which is the diagonal. So, the major axis length is 2a, which is the distance between E and G, which is √(x^2 + y^2). So, 2a = √(x^2 + y^2). Therefore, a = √(x^2 + y^2)/2.But then, the distance between the foci is 2c, which is the distance between F and H, which is also √(x^2 + y^2). So, 2c = √(x^2 + y^2). Therefore, c = √(x^2 + y^2)/2, which is equal to a. But in an ellipse, c < a, so this would imply that c = a, which is only possible if the ellipse is a line segment, which is degenerate. So, that can't be.Hmm, so my assumption that the major axis is the diagonal must be wrong. Maybe the major axis is not the diagonal, but something else. Let me think.Alternatively, perhaps the major axis is along one of the sides of the rectangle. Let's say the major axis is along the length x. Then, the major axis length would be 2a, and the distance between the foci would be 2c, which is along the same axis. But in this case, the foci are at F and H, which are opposite corners, so the distance between them is the diagonal. So, that would mean that 2c = √(x^2 + y^2). So, c = √(x^2 + y^2)/2.But if the major axis is along the length x, then the major axis length is 2a, which would be equal to x + something. Wait, no, the major axis is the longest diameter of the ellipse, so if the major axis is along x, then the ellipse is stretched along x, and the distance from the center to each focus is c, along the major axis.But in this case, the foci are at F and H, which are opposite corners, so the line connecting F and H is the diagonal, not along the major axis. So, that suggests that the major axis is not aligned with the sides of the rectangle, but at some angle.Wait, maybe the major axis is the line connecting E and G, which is the diagonal, but then as I thought before, that would imply c = a, which is degenerate. So, that can't be.Hmm, this is confusing. Maybe I need to approach this differently.Let me recall that in an ellipse, the sum of the distances from any point on the ellipse to the two foci is constant and equal to 2a. So, for point E, which is on the ellipse, the sum of distances to F and H is 2a. Similarly, for point G, the sum of distances to F and H is also 2a.So, let's compute these distances.For point E, which is a corner of the rectangle, the distance to F is one side, say x, and the distance to H is the diagonal, √(x^2 + y^2). So, x + √(x^2 + y^2) = 2a.Similarly, for point G, the distance to F is the diagonal √(x^2 + y^2), and the distance to H is y. So, √(x^2 + y^2) + y = 2a.So, we have two equations:1. x + √(x^2 + y^2) = 2a2. y + √(x^2 + y^2) = 2aSubtracting these two equations, we get x - y = 0, so x = y. Therefore, the rectangle must be a square.Wait, that's interesting. So, the rectangle is actually a square with sides x = y. Then, the area is x^2 = 2450, so x = √2450 = 35√2, as I calculated earlier.But then, the perimeter would be 4x = 4*35√2 = 140√2. But let me check if this makes sense with the ellipse.If the rectangle is a square, then the diagonal is x√2 = 35√2 * √2 = 35*2 = 70. So, the distance between F and H is 70, which is 2c. So, c = 35.The area of the ellipse is πab = 2450π, so ab = 2450.Also, in an ellipse, c^2 = a^2 - b^2. We have c = 35, so 35^2 = a^2 - b^2 => 1225 = a^2 - b^2.But we also have ab = 2450. So, we have two equations:1. a^2 - b^2 = 12252. ab = 2450Let me solve these equations. From equation 2, b = 2450/a. Substitute into equation 1:a^2 - (2450/a)^2 = 1225Multiply both sides by a^2:a^4 - 2450^2 = 1225a^2Rearrange:a^4 - 1225a^2 - 2450^2 = 0Let me let z = a^2, then:z^2 - 1225z - 2450^2 = 0This is a quadratic in z:z^2 - 1225z - (2450)^2 = 0Using quadratic formula:z = [1225 ± √(1225^2 + 4*2450^2)] / 2Compute discriminant:D = 1225^2 + 4*(2450)^2Note that 2450 = 2*1225, so 2450^2 = 4*1225^2Thus, D = 1225^2 + 4*(4*1225^2) = 1225^2 + 16*1225^2 = 17*1225^2Therefore, √D = √(17)*1225So, z = [1225 ± 1225√17]/2Since z = a^2 must be positive, we take the positive root:z = [1225 + 1225√17]/2 = 1225(1 + √17)/2Thus, a^2 = 1225(1 + √17)/2Therefore, a = √[1225(1 + √17)/2] = √[1225/2 * (1 + √17)] = (35/√2) * √(1 + √17)Hmm, this seems complicated. Maybe I made a mistake earlier.Wait, if the rectangle is a square, then the ellipse passing through E and G with foci at F and H would have its major axis along the diagonal. But earlier, I thought that would imply c = a, which is degenerate, but maybe not.Wait, let's think again. If the major axis is the diagonal, then 2a = diagonal = 70, so a = 35. Then, the distance between foci is 2c = 70, so c = 35. But then, c = a, which would mean b = 0, which is degenerate. So, that can't be.Therefore, my earlier conclusion that the rectangle is a square must be wrong. So, where did I go wrong?Wait, I had two equations:1. x + √(x^2 + y^2) = 2a2. y + √(x^2 + y^2) = 2aSubtracting them gives x - y = 0, so x = y. But maybe that's not the case because the ellipse is not necessarily symmetric in that way. Wait, no, the ellipse has foci at F and H, which are opposite corners, so the ellipse is symmetric with respect to the center of the rectangle. Therefore, if the rectangle is not a square, the ellipse would have different properties.Wait, perhaps I need to consider that the major axis is not along the diagonal, but at some angle, making the ellipse rotated with respect to the rectangle.Alternatively, maybe the major axis is not aligned with the rectangle's sides or diagonal, but somewhere in between.This is getting complicated. Maybe I need to use coordinate geometry.Let me place the rectangle EFGH on a coordinate system. Let me assume that E is at (0,0), F is at (x,0), G is at (x,y), and H is at (0,y). So, the rectangle has vertices at E(0,0), F(x,0), G(x,y), H(0,y).The foci of the ellipse are at F(x,0) and H(0,y). The ellipse passes through E(0,0) and G(x,y).So, the ellipse has foci at (x,0) and (0,y), and passes through (0,0) and (x,y).Let me recall the general equation of an ellipse given two foci and a point on the ellipse. The sum of distances from any point on the ellipse to the two foci is constant and equal to 2a.So, for point E(0,0), the sum of distances to F(x,0) and H(0,y) is:Distance from E to F: √[(x - 0)^2 + (0 - 0)^2] = xDistance from E to H: √[(0 - 0)^2 + (y - 0)^2] = ySo, sum = x + y = 2aSimilarly, for point G(x,y), the sum of distances to F(x,0) and H(0,y) is:Distance from G to F: √[(x - x)^2 + (0 - y)^2] = yDistance from G to H: √[(0 - x)^2 + (y - y)^2] = xSo, sum = y + x = 2aTherefore, both points E and G give the same sum, which is x + y = 2a.So, 2a = x + y.Now, the distance between the foci is the distance between F(x,0) and H(0,y), which is √[(x - 0)^2 + (0 - y)^2] = √(x^2 + y^2). This distance is equal to 2c, where c is the distance from the center to each focus.So, 2c = √(x^2 + y^2), so c = √(x^2 + y^2)/2.In an ellipse, we have the relationship c^2 = a^2 - b^2, where b is the semi-minor axis.So, (√(x^2 + y^2)/2)^2 = a^2 - b^2Which simplifies to (x^2 + y^2)/4 = a^2 - b^2We also know that the area of the ellipse is πab = 2450π, so ab = 2450.We have:1. 2a = x + y2. (x^2 + y^2)/4 = a^2 - b^23. ab = 2450We also know that the area of the rectangle is xy = 2450.So, we have four equations:1. 2a = x + y2. (x^2 + y^2)/4 = a^2 - b^23. ab = 24504. xy = 2450Let me see how to solve these.From equation 1: x + y = 2aFrom equation 4: xy = 2450We can express x and y in terms of a and some other variable, but maybe it's better to express x^2 + y^2 in terms of a.We know that (x + y)^2 = x^2 + 2xy + y^2So, x^2 + y^2 = (x + y)^2 - 2xy = (2a)^2 - 2*2450 = 4a^2 - 4900Therefore, equation 2 becomes:(4a^2 - 4900)/4 = a^2 - b^2Simplify:(4a^2 - 4900)/4 = a^2 - b^2Multiply numerator:(4a^2)/4 - 4900/4 = a^2 - b^2Simplify:a^2 - 1225 = a^2 - b^2Subtract a^2 from both sides:-1225 = -b^2Multiply both sides by -1:1225 = b^2So, b = √1225 = 35Now, from equation 3: ab = 2450We have b = 35, so a = 2450 / 35 = 70So, a = 70, b = 35Now, from equation 1: x + y = 2a = 140From equation 4: xy = 2450So, we have x + y = 140 and xy = 2450We can solve for x and y.Let me set up the quadratic equation:Let x and y be the roots of t^2 - (x + y)t + xy = 0 => t^2 - 140t + 2450 = 0Solve for t:t = [140 ± √(140^2 - 4*1*2450)] / 2Compute discriminant:D = 19600 - 9800 = 9800√D = √9800 = √(100*98) = 10√98 = 10*7√2 = 70√2So, t = [140 ± 70√2]/2 = 70 ± 35√2Therefore, x = 70 + 35√2 and y = 70 - 35√2, or vice versa.But since x and y are lengths, they must be positive. Let's check:70 + 35√2 ≈ 70 + 35*1.414 ≈ 70 + 49.49 ≈ 119.4970 - 35√2 ≈ 70 - 49.49 ≈ 20.51Both are positive, so that's fine.Therefore, the sides of the rectangle are 70 + 35√2 and 70 - 35√2.Now, the perimeter of the rectangle is 2(x + y) = 2*140 = 280.Wait, but that seems too simple. Wait, x + y = 140, so perimeter is 2(x + y) = 280.But let me double-check.Wait, x + y = 140, so perimeter is 2(x + y) = 280. Yes, that's correct.But let me make sure that the ellipse parameters make sense.We have a = 70, b = 35, c = √(a^2 - b^2) = √(4900 - 1225) = √3675 = √(25*147) = 5√147 = 5*√(49*3) = 5*7√3 = 35√3So, c = 35√3But earlier, we had c = √(x^2 + y^2)/2Compute √(x^2 + y^2):x = 70 + 35√2, y = 70 - 35√2x^2 = (70 + 35√2)^2 = 70^2 + 2*70*35√2 + (35√2)^2 = 4900 + 4900√2 + 2450*2 = 4900 + 4900√2 + 4900 = 9800 + 4900√2Similarly, y^2 = (70 - 35√2)^2 = 70^2 - 2*70*35√2 + (35√2)^2 = 4900 - 4900√2 + 4900 = 9800 - 4900√2So, x^2 + y^2 = (9800 + 4900√2) + (9800 - 4900√2) = 19600Therefore, √(x^2 + y^2) = √19600 = 140So, c = 140 / 2 = 70Wait, but earlier, I had c = 35√3 ≈ 60.62, but here, c = 70. That's a contradiction.Wait, what's wrong here?Wait, no, I think I made a mistake in calculating c.Wait, c is the distance from the center to each focus. The distance between the foci is 2c, which is equal to √(x^2 + y^2) = 140. So, 2c = 140, so c = 70.But earlier, from the ellipse parameters, c = √(a^2 - b^2) = √(70^2 - 35^2) = √(4900 - 1225) = √3675 ≈ 60.62But this contradicts c = 70.So, something is wrong here.Wait, let's go back.We have:From the ellipse:c = √(a^2 - b^2) = √(70^2 - 35^2) = √(4900 - 1225) = √3675 ≈ 60.62But from the rectangle:Distance between foci (F and H) is √(x^2 + y^2) = 140, so 2c = 140, so c = 70.But 70 ≠ √3675 ≈ 60.62This is a contradiction. So, my earlier steps must have an error.Wait, where did I go wrong?Let me retrace.We have:1. 2a = x + y2. (x^2 + y^2)/4 = a^2 - b^23. ab = 24504. xy = 2450From 1: x + y = 2aFrom 4: xy = 2450From 2: (x^2 + y^2)/4 = a^2 - b^2We know that x^2 + y^2 = (x + y)^2 - 2xy = (2a)^2 - 2*2450 = 4a^2 - 4900So, equation 2 becomes:(4a^2 - 4900)/4 = a^2 - b^2Simplify:a^2 - 1225 = a^2 - b^2Therefore, -1225 = -b^2 => b^2 = 1225 => b = 35From equation 3: ab = 2450 => a = 2450 / 35 = 70So, a = 70, b = 35Then, from equation 1: x + y = 140From equation 4: xy = 2450So, solving for x and y, we get x = 70 + 35√2, y = 70 - 35√2Then, x^2 + y^2 = 19600, so √(x^2 + y^2) = 140, which is 2c, so c = 70But in the ellipse, c = √(a^2 - b^2) = √(4900 - 1225) = √3675 ≈ 60.62This is a contradiction because c should be 70, but according to the ellipse formula, it's √3675.So, where is the mistake?Wait, perhaps the assumption that the sum of distances from E to F and H is 2a is incorrect.Wait, no, in an ellipse, the sum of distances from any point on the ellipse to the two foci is 2a. So, for point E, which is on the ellipse, the sum of distances to F and H is 2a.But in our case, the distance from E to F is x, and from E to H is √(x^2 + y^2). So, x + √(x^2 + y^2) = 2aSimilarly, for point G, the distance to F is √(x^2 + y^2), and to H is y, so √(x^2 + y^2) + y = 2aTherefore, x + √(x^2 + y^2) = y + √(x^2 + y^2) => x = yBut earlier, we saw that leads to a contradiction because the ellipse parameters don't match.Wait, but if x = y, then the rectangle is a square, and we saw that leads to c = a, which is degenerate.So, perhaps the initial assumption that the ellipse passes through E and G with foci at F and H is only possible if the rectangle is a square, but that leads to a degenerate ellipse, which is not the case here.Alternatively, maybe the ellipse is not centered at the center of the rectangle, but that complicates things.Wait, in a rectangle, the center is the intersection of the diagonals, which is also the midpoint between E and G, and between F and H. So, the center of the ellipse should be at this point.But in our earlier calculation, we assumed that the ellipse is centered at the center of the rectangle, which is correct.Wait, but if the ellipse is centered at the center of the rectangle, then the distance from the center to each focus is c, and the distance between the foci is 2c.But in our case, the distance between F and H is √(x^2 + y^2), which is 2c.But we also have that c = √(a^2 - b^2)So, in our case, 2c = √(x^2 + y^2) = 140, so c = 70But from the ellipse, c = √(a^2 - b^2) = √(70^2 - 35^2) = √3675 ≈ 60.62Which is not equal to 70.So, this is a contradiction.Therefore, my earlier approach must be wrong.Wait, maybe I need to consider that the major axis is not aligned with the rectangle's sides or diagonal, but at some angle, making the ellipse rotated.This complicates things because then the standard ellipse equations don't apply directly.Alternatively, perhaps the major axis is along the line connecting E and G, which is the diagonal, but then as before, that would imply c = a, which is degenerate.Wait, but in our case, a = 70, c = 70, so b = 0, which is degenerate. So, that can't be.Therefore, perhaps the initial assumption that the ellipse passes through E and G with foci at F and H is only possible if the rectangle is a square, but that leads to a degenerate ellipse, which contradicts the given area.Therefore, maybe the problem is designed such that the rectangle is a square, and despite the degenerate ellipse, we proceed with the calculations.But in that case, the perimeter would be 140√2, but earlier, we saw that leads to a degenerate ellipse.Alternatively, perhaps the problem is designed such that the ellipse is not degenerate, and the rectangle is not a square, but our earlier approach is missing something.Wait, perhaps the major axis is not the line connecting E and G, but something else.Wait, let me think differently.Let me denote the center of the ellipse as the center of the rectangle, which is at (x/2, y/2).The distance from the center to each focus is c, so the foci are located at (x/2 ± c_x, y/2 ± c_y). But in our case, the foci are at F(x,0) and H(0,y). So, the coordinates of the foci are (x,0) and (0,y).So, the center is at (x/2, y/2), and the foci are at (x,0) and (0,y). Therefore, the vector from the center to each focus is (x/2, -y/2) and (-x/2, y/2). So, the distance from the center to each focus is √[(x/2)^2 + (y/2)^2] = √(x^2 + y^2)/2 = cSo, c = √(x^2 + y^2)/2But in the ellipse, c = √(a^2 - b^2)So, √(x^2 + y^2)/2 = √(a^2 - b^2)We also have that the sum of distances from E(0,0) to F(x,0) and H(0,y) is 2a.So, distance from E to F is x, distance from E to H is y, so x + y = 2aSimilarly, for point G(x,y), distance to F is y, distance to H is x, so y + x = 2aSo, same as before, x + y = 2aWe also have that the area of the ellipse is πab = 2450π, so ab = 2450And the area of the rectangle is xy = 2450So, we have:1. x + y = 2a2. xy = 24503. ab = 24504. √(x^2 + y^2)/2 = √(a^2 - b^2)Let me try to express everything in terms of a and b.From equation 1: x + y = 2aFrom equation 2: xy = 2450From equation 3: ab = 2450 => b = 2450/aFrom equation 4: √(x^2 + y^2)/2 = √(a^2 - b^2)Let me compute x^2 + y^2.From equation 1 and 2:(x + y)^2 = x^2 + 2xy + y^2 => (2a)^2 = x^2 + 2*2450 + y^2 => 4a^2 = x^2 + y^2 + 4900 => x^2 + y^2 = 4a^2 - 4900So, equation 4 becomes:√(4a^2 - 4900)/2 = √(a^2 - b^2)Square both sides:(4a^2 - 4900)/4 = a^2 - b^2Simplify:(a^2 - 1225) = a^2 - b^2So, -1225 = -b^2 => b^2 = 1225 => b = 35From equation 3: ab = 2450 => a = 2450 / 35 = 70So, a = 70, b = 35From equation 1: x + y = 140From equation 2: xy = 2450So, solving for x and y:x and y are roots of t^2 - 140t + 2450 = 0Solutions:t = [140 ± √(140^2 - 4*2450)] / 2 = [140 ± √(19600 - 9800)] / 2 = [140 ± √9800]/2 = [140 ± 70√2]/2 = 70 ± 35√2So, x = 70 + 35√2, y = 70 - 35√2Now, compute √(x^2 + y^2):x^2 = (70 + 35√2)^2 = 4900 + 4900√2 + 2450 = 7350 + 4900√2y^2 = (70 - 35√2)^2 = 4900 - 4900√2 + 2450 = 7350 - 4900√2So, x^2 + y^2 = 7350 + 4900√2 + 7350 - 4900√2 = 14700Wait, earlier I thought x^2 + y^2 was 19600, but that was a miscalculation.Wait, let me recalculate x^2 and y^2 correctly.x = 70 + 35√2x^2 = (70)^2 + 2*70*35√2 + (35√2)^2 = 4900 + 4900√2 + 35^2*2 = 4900 + 4900√2 + 1225*2 = 4900 + 4900√2 + 2450 = 7350 + 4900√2Similarly, y = 70 - 35√2y^2 = (70)^2 - 2*70*35√2 + (35√2)^2 = 4900 - 4900√2 + 2450 = 7350 - 4900√2So, x^2 + y^2 = (7350 + 4900√2) + (7350 - 4900√2) = 14700Therefore, √(x^2 + y^2) = √14700 = √(100*147) = 10√147 = 10*√(49*3) = 10*7√3 = 70√3So, √(x^2 + y^2) = 70√3Therefore, equation 4: √(x^2 + y^2)/2 = 70√3 / 2 = 35√3But from the ellipse, √(a^2 - b^2) = √(70^2 - 35^2) = √(4900 - 1225) = √3675 = √(25*147) = 5√147 = 5*7√3 = 35√3So, √(a^2 - b^2) = 35√3, which matches √(x^2 + y^2)/2 = 35√3Therefore, there was a miscalculation earlier when I thought √(x^2 + y^2) was 140. Actually, it's 70√3, which is approximately 121.24, not 140.So, the distance between the foci is 2c = 70√3, so c = 35√3Which matches with c = √(a^2 - b^2) = 35√3Therefore, everything checks out.So, the sides of the rectangle are x = 70 + 35√2 and y = 70 - 35√2Therefore, the perimeter is 2(x + y) = 2*140 = 280Wait, but earlier, I thought x + y = 140, so perimeter is 280.But let me confirm:x = 70 + 35√2 ≈ 70 + 49.497 ≈ 119.497y = 70 - 35√2 ≈ 70 - 49.497 ≈ 20.503So, x + y ≈ 119.497 + 20.503 ≈ 140Therefore, perimeter is 2*(x + y) = 280But wait, the problem asks for the perimeter of rectangle EFGH, which is 2(x + y) = 280But let me check if this makes sense with the ellipse.We have a = 70, b = 35, c = 35√3So, the major axis is 2a = 140, minor axis is 2b = 70The distance between foci is 2c = 70√3 ≈ 121.24Which is less than 2a = 140, which is correct because in an ellipse, 2c < 2aSo, everything checks out.Therefore, the perimeter of the rectangle is 280.But wait, in the initial problem, the area of the ellipse is 2450π, which is πab = 2450π, so ab = 2450With a = 70, b = 35, ab = 2450, which matches.And the area of the rectangle is xy = 2450, which is also satisfied.Therefore, the perimeter is 280.But wait, in the initial problem, the user wrote:"Rectangle EFGH has an area of 2450. An ellipse with an area of 2450π passes through the points E and G, and has foci at F and H. Determine the perimeter of rectangle EFGH."So, according to my calculations, the perimeter is 280.But in the initial assistant's answer, it was 120√17, which is approximately 120*4.123 = 494.76, which is much larger than 280.So, perhaps the initial assistant made a mistake.Wait, let me check the initial assistant's steps.They said:Let 2a and 2b be the lengths of the major and minor axes of the ellipse, respectively, and let the dimensions of the rectangle be x and y. Then, x+y equals the sum of the distances from the foci to point E on the ellipse, which is 2a, so x+y = 2a. The length of a diagonal of the rectangle, which is also the distance between the foci of the ellipse, is √(x^2+y^2) and equals 2√(a^2 - b^2). Therefore, we have the equation x^2+y^2 = 4(a^2 - b^2).Given that the area of the rectangle is 2450 = xy, we use the equation:2450 = (1/2)[(x+y)^2 - (x^2+y^2)] = 2a^2 - 2b^2Solving for b^2, we find:b^2 = a^2 - 1225Given the ellipse's area:2450π = πab = πa√(a^2 - 1225)From which we find:2450 = a√(a^2 - 1225)Let's solve for a:a = √(2450 + 1225) = √3675Hence, a = 15√17, and b = √(15√17^2 - 1225) = √(3675 - 1225) = 15√34 - 17 = 15√17, ensuring b^2 = 1225 and a = 30√17.The perimeter of the rectangle is 2(x+y) = 4a = 4 × 30√17 = 120√17.Wait, this is different from my approach. They assumed that x + y = 2a, and that the diagonal is 2√(a^2 - b^2). But in reality, the distance between the foci is 2c = 2√(a^2 - b^2), which is equal to the diagonal √(x^2 + y^2). So, they set √(x^2 + y^2) = 2√(a^2 - b^2), which is correct.But then, they used the equation 2450 = (1/2)[(x+y)^2 - (x^2 + y^2)] = 2a^2 - 2b^2Wait, let me see:(x + y)^2 = x^2 + 2xy + y^2So, (x + y)^2 - (x^2 + y^2) = 2xy = 2*2450 = 4900Therefore, (1/2)[(x + y)^2 - (x^2 + y^2)] = (1/2)*4900 = 2450Which equals 2a^2 - 2b^2, as they said.So, 2a^2 - 2b^2 = 2450 => a^2 - b^2 = 1225Which is correct.Then, they have ab = 2450So, they have:1. a^2 - b^2 = 12252. ab = 2450They solve for a and b.From equation 2: b = 2450/aSubstitute into equation 1:a^2 - (2450/a)^2 = 1225Multiply both sides by a^2:a^4 - 2450^2 = 1225a^2Which is a quartic equation.They then let a^2 = z:z^2 - 1225z - 2450^2 = 0Solve using quadratic formula:z = [1225 ± √(1225^2 + 4*2450^2)] / 2Compute discriminant:D = 1225^2 + 4*(2450)^2 = 1225^2 + 4*(2*1225)^2 = 1225^2 + 4*4*1225^2 = 1225^2*(1 + 16) = 17*1225^2So, √D = 1225√17Thus, z = [1225 + 1225√17]/2 = 1225(1 + √17)/2Therefore, a^2 = 1225(1 + √17)/2So, a = √[1225(1 + √17)/2] = √[1225/2 * (1 + √17)] = (35/√2) * √(1 + √17)But they simplified it as a = 15√17, which is incorrect.Wait, let me compute 15√17:15√17 ≈ 15*4.123 ≈ 61.845But from our earlier correct calculation, a = 70So, their value of a is incorrect.They made a mistake in solving the quadratic equation.Wait, let me see:They had z = [1225 + 1225√17]/2So, z = 1225(1 + √17)/2Therefore, a^2 = 1225(1 + √17)/2So, a = √[1225(1 + √17)/2] = (35)√[(1 + √17)/2]But they wrote a = 15√17, which is not correct.So, their mistake was in solving for a.Therefore, their conclusion that a = 30√17 is wrong.In reality, a = 70, as we found earlier.Therefore, the correct perimeter is 280.But in the initial problem, the user wrote:"Rectangle EFGH has an area of 2450. An ellipse with an area of 2450π passes through the points E and G, and has foci at F and H. Determine the perimeter of rectangle EFGH."So, the correct answer is 280.But in the initial assistant's answer, they got 120√17, which is approximately 494.76, which is incorrect.Therefore, the correct perimeter is 280.

💡Okay, so I have this problem here about an arithmetic sequence and a geometric sequence. Let me try to parse it step by step.First, the arithmetic sequence is denoted as {a_n} with a common difference d, and d is not zero. The geometric sequence is {b_n} with a positive rational common ratio q. Given that a₁ = d and b₁ = d². Then, the expression (a₁² + a₂² + a₃²) divided by (b₁ + b₂ + b₃) is a positive integer. We need to find the value of q from the options given.Alright, let me break this down.First, let's recall what an arithmetic sequence is. Each term is the previous term plus a common difference. So, if a₁ is d, then a₂ is a₁ + d, which is d + d = 2d. Similarly, a₃ would be a₂ + d = 2d + d = 3d. So, the first three terms of the arithmetic sequence are d, 2d, and 3d.Now, for the geometric sequence. Each term is the previous term multiplied by a common ratio q. So, if b₁ is d², then b₂ is b₁ * q = d² * q, and b₃ is b₂ * q = d² * q². So, the first three terms of the geometric sequence are d², d²q, and d²q².Now, the expression we need to evaluate is (a₁² + a₂² + a₃²) divided by (b₁ + b₂ + b₃). Let's compute the numerator and the denominator separately.Starting with the numerator: a₁² + a₂² + a₃².We already have a₁ = d, a₂ = 2d, a₃ = 3d.So, a₁² = d², a₂² = (2d)² = 4d², a₃² = (3d)² = 9d².Adding these up: d² + 4d² + 9d² = 14d². So, the numerator is 14d².Now, the denominator: b₁ + b₂ + b₃.We have b₁ = d², b₂ = d²q, b₃ = d²q².Adding these up: d² + d²q + d²q². We can factor out d²: d²(1 + q + q²). So, the denominator is d²(1 + q + q²).Therefore, the entire expression becomes:(14d²) / [d²(1 + q + q²)].We can cancel out d² from numerator and denominator, since d ≠ 0. So, we get 14 / (1 + q + q²).The problem states that this expression is a positive integer. So, 14 divided by (1 + q + q²) is an integer. Let's denote this integer as k, so:14 / (1 + q + q²) = k, where k is a positive integer.Therefore, 1 + q + q² = 14 / k.Since q is a positive rational number, let's denote q as a fraction in its simplest form. Let me think about how to approach this.Given that q is rational, let me write q as m/n where m and n are positive integers with no common factors.So, q = m/n, with m and n positive integers, gcd(m, n) = 1.Then, 1 + q + q² = 1 + m/n + (m/n)² = 1 + m/n + m²/n².Let me write this as a single fraction:1 + m/n + m²/n² = (n² + mn + m²) / n².So, 1 + q + q² = (n² + mn + m²) / n².Therefore, 14 / (1 + q + q²) = 14 * (n²) / (n² + mn + m²).We know this equals k, a positive integer. So,14n² / (n² + mn + m²) = k.Therefore, (n² + mn + m²) must divide 14n².So, (n² + mn + m²) divides 14n².Given that m and n are coprime, let's see if we can find integers m and n such that (n² + mn + m²) divides 14n².Alternatively, since 14n² / (n² + mn + m²) is an integer, let's denote t = (n² + mn + m²). Then, t divides 14n².But t = n² + mn + m².Since m and n are coprime, let's see if we can find possible values of m and n.Alternatively, perhaps it's easier to consider possible integer values of k, since k must be a positive integer divisor of 14.So, let's list the positive integer divisors of 14: 1, 2, 7, 14.So, k can be 1, 2, 7, or 14.Therefore, 1 + q + q² = 14 / k.So, for each possible k, we can compute 1 + q + q² and solve for q.Let me try each possible k:1. If k = 1, then 1 + q + q² = 14 / 1 = 14.So, q² + q + 1 = 14.Then, q² + q - 13 = 0.Solving this quadratic equation: q = [-1 ± sqrt(1 + 52)] / 2 = [-1 ± sqrt(53)] / 2.But sqrt(53) is irrational, so q would be irrational, which contradicts the condition that q is a positive rational number. So, k = 1 is invalid.2. If k = 2, then 1 + q + q² = 14 / 2 = 7.So, q² + q + 1 = 7.Then, q² + q - 6 = 0.Solving this quadratic equation: q = [-1 ± sqrt(1 + 24)] / 2 = [-1 ± 5] / 2.Since q is positive, we take the positive root: q = (-1 + 5)/2 = 4/2 = 2.So, q = 2 is a solution. Let's check if this is valid.Yes, q = 2 is a positive rational number. So, k = 2 is valid.3. If k = 7, then 1 + q + q² = 14 / 7 = 2.So, q² + q + 1 = 2.Then, q² + q - 1 = 0.Solving this quadratic equation: q = [-1 ± sqrt(1 + 4)] / 2 = [-1 ± sqrt(5)] / 2.Again, sqrt(5) is irrational, so q would be irrational, which is not allowed. So, k = 7 is invalid.4. If k = 14, then 1 + q + q² = 14 / 14 = 1.So, q² + q + 1 = 1.Then, q² + q = 0.Solving this: q(q + 1) = 0.Solutions are q = 0 or q = -1.But q is a positive rational number, so neither 0 nor -1 are acceptable. Therefore, k = 14 is invalid.So, the only valid value is k = 2, which gives q = 2.Wait, but the options include 2 and 1/2. Did I miss something?Wait, let me think again. Maybe I need to consider that q could be a fraction, so perhaps m/n where m < n.Wait, when I considered k = 2, I got q = 2. But maybe if I set k = 8, which is not a divisor of 14, but perhaps I made a mistake in the possible k values.Wait, no, k must be a positive integer such that 14 / k is an integer. So, k must be a divisor of 14. So, k can only be 1, 2, 7, 14.But in the options, there's a choice D: 2 or 1/2. So, maybe I missed another possible k.Wait, perhaps I made a mistake in assuming that k must be a positive integer divisor of 14. Let me think again.Wait, 14 / (1 + q + q²) = k, where k is a positive integer. So, 1 + q + q² must be a divisor of 14, but not necessarily an integer divisor. Wait, no, because 1 + q + q² is a rational number since q is rational. So, 14 divided by a rational number is an integer, so 1 + q + q² must be a rational number that divides 14.But since q is rational, let me write q = m/n in lowest terms, so m and n are coprime positive integers.Then, 1 + q + q² = 1 + m/n + m²/n² = (n² + mn + m²)/n².So, 14 divided by (n² + mn + m²)/n² is equal to 14n² / (n² + mn + m²) = k, which is an integer.So, 14n² must be divisible by (n² + mn + m²). So, (n² + mn + m²) divides 14n².Since m and n are coprime, let's see if we can find m and n such that n² + mn + m² divides 14n².Let me denote t = n² + mn + m². So, t divides 14n².Since t divides 14n², and t = n² + mn + m², which is greater than n² (since m and n are positive integers), unless m = 0, which is not allowed because q is positive.So, t must be a divisor of 14n², and t > n².But t = n² + mn + m².Let me consider possible small values of m and n.Case 1: m = 1, n = 1.Then, t = 1 + 1 + 1 = 3.So, 14n² / t = 14*1 / 3 ≈ 4.666, which is not integer.Case 2: m = 1, n = 2.t = 4 + 2 + 1 = 7.14n² / t = 14*4 / 7 = 56 / 7 = 8, which is integer. So, k = 8.So, q = m/n = 1/2.So, when m = 1, n = 2, we get q = 1/2, and k = 8.So, that's another solution.Similarly, let's check m = 2, n = 1.t = 1 + 2 + 4 = 7.14n² / t = 14*1 /7 = 2, which is integer. So, k = 2, and q = 2/1 = 2.So, that's the solution we found earlier.So, in total, we have two possible values for q: 2 and 1/2.Therefore, the correct answer is D: 2 or 1/2.Wait, but in my initial approach, I only considered k as divisors of 14, but I missed that k can be 8, which is not a divisor of 14, but when considering q as a fraction, it's possible.So, the key was to consider q as a fraction m/n and find m and n such that 14n² is divisible by (n² + mn + m²). This led us to two solutions: q = 2 and q = 1/2.Therefore, the answer is D.

💡Okay, so I've got this problem here about Ray, Tom, and Amy driving their cars. Each of them drives 120 miles, but their cars have different fuel efficiencies. Ray's car gets 50 miles per gallon, Tom's gets 20 miles per gallon, and Amy's gets 40 miles per gallon. The question is asking for the combined rate of miles per gallon for all three cars together. Hmm, okay, let's break this down step by step.First, I need to understand what a combined rate of miles per gallon means. I think it refers to the total miles driven by all three cars divided by the total gallons of gasoline they used. So, I need to calculate how much gasoline each person used for their 120-mile trip and then add them all up. After that, I'll add up all the miles they drove and divide that by the total gallons used. That should give me the combined miles per gallon.Alright, let's start with Ray. Ray's car averages 50 miles per gallon. So, if he drives 120 miles, how many gallons does he use? I can calculate that by dividing the miles driven by the miles per gallon. So, 120 miles divided by 50 miles per gallon. Let me do that: 120 ÷ 50 equals 2.4 gallons. Okay, so Ray uses 2.4 gallons of gas for his trip.Next, Tom's car averages 20 miles per gallon. He also drives 120 miles. So, how many gallons does Tom use? Again, I'll divide the miles driven by the miles per gallon. That's 120 ÷ 20, which equals 6 gallons. So, Tom uses 6 gallons of gas for his trip.Now, Amy's car averages 40 miles per gallon. She also drives 120 miles. Let's find out how many gallons she uses. Dividing 120 by 40 gives me 3 gallons. So, Amy uses 3 gallons of gas for her trip.Okay, so now I have the gallons used by each person: Ray used 2.4 gallons, Tom used 6 gallons, and Amy used 3 gallons. To find the total gallons used by all three, I need to add these together. Let me add them up: 2.4 + 6 + 3. Hmm, 2.4 plus 6 is 8.4, and then adding 3 gives me 11.4 gallons total.Now, the total miles driven by all three is straightforward since each drove 120 miles. So, 120 miles for Ray, 120 for Tom, and 120 for Amy. Adding those together: 120 + 120 + 120 equals 360 miles total.Alright, so now I have 360 miles driven in total and 11.4 gallons used in total. To find the combined miles per gallon, I divide the total miles by the total gallons. So, 360 ÷ 11.4. Let me calculate that.First, I can simplify this division. 360 divided by 11.4. To make it easier, I can multiply both numerator and denominator by 10 to eliminate the decimal. That gives me 3600 ÷ 114. Now, let's see how many times 114 goes into 3600.114 times 30 is 3420. Subtracting that from 3600 gives me 180. Then, 114 goes into 180 once, with a remainder of 66. So, 30 + 1 is 31, and there's a remainder. To get a decimal, I can add a decimal point and continue dividing. 114 goes into 660 five times (since 114 x 5 is 570), leaving a remainder of 90. Then, 114 goes into 900 seven times (114 x 7 is 798), leaving a remainder of 102. Hmm, this is getting a bit messy, but I can see that 3600 ÷ 114 is approximately 31.5789.Rounding that to two decimal places, it's about 31.58 miles per gallon. Looking back at the answer choices, I see that option C is 31.55, which is very close to my calculation. I think that's the answer they're looking for.Wait, let me double-check my calculations to make sure I didn't make a mistake. So, Ray uses 2.4 gallons, Tom uses 6 gallons, Amy uses 3 gallons. Total gallons: 2.4 + 6 + 3 = 11.4 gallons. Total miles: 120 x 3 = 360 miles. 360 ÷ 11.4 = approximately 31.58. Yeah, that seems right.I guess the slight difference between 31.58 and 31.55 is probably due to rounding during the division process. So, 31.55 is the closest answer choice, so I'll go with that.**Final Answer**boxed{C}

💡First, I need to find the dot product of vectors (overrightarrow{a}) and (overrightarrow{b}). The dot product is calculated by multiplying the corresponding components of the vectors and then summing those products.Next, I'll calculate the magnitudes of both vectors. The magnitude of a vector is found by taking the square root of the sum of the squares of its components.With the dot product and the magnitudes known, I can use the formula for the cosine of the angle between two vectors. This formula involves dividing the dot product by the product of the magnitudes of the two vectors.After finding the cosine of the angle, I'll determine the angle itself by identifying which standard angle has that cosine value. This will give me the angle between the two vectors.

💡Alright, so I have this problem about a right prism, specifically ABC-A₁B₁C₁. The given information is that AB is 3 units, AC is 4 units, and AB is perpendicular to AC. Also, the height of the prism, AA₁, is 2 units. I need to find the sum of the surface areas of the inscribed sphere and the circumscribed sphere of the prism.Hmm, okay. Let me start by visualizing the prism. Since it's a right prism, the sides are rectangles, and the bases are congruent triangles. The base triangle ABC has sides AB=3 and AC=4, and since AB is perpendicular to AC, triangle ABC is a right-angled triangle. So, ABC is a right triangle with legs 3 and 4. That means the hypotenuse BC can be found using the Pythagorean theorem.Calculating BC: BC = √(AB² + AC²) = √(3² + 4²) = √(9 + 16) = √25 = 5. Okay, so the sides of triangle ABC are 3, 4, and 5. That makes it a 3-4-5 right triangle, which is a common Pythagorean triple.Now, the prism has a height of 2 units, so each of the vertices A, B, C is connected to A₁, B₁, C₁ respectively, each 2 units above the base.I need to find the surface areas of both the inscribed sphere and the circumscribed sphere of this prism. Then, sum them up.First, let me think about the circumscribed sphere. A circumscribed sphere (circum sphere) of a prism would pass through all the vertices of the prism. Since it's a right prism, the circum sphere's center should be equidistant from all the vertices.To find the radius of the circum sphere, I can consider the prism as a rectangular box. Wait, but it's a prism with a right triangle base, so it's not a rectangular box, but maybe I can still use some similar reasoning.In a rectangular box, the space diagonal is the diameter of the circumscribed sphere. The space diagonal can be found using the formula √(length² + width² + height²). In this case, the prism isn't a rectangular box, but the base is a right triangle. Maybe I can still use the same approach by considering the prism as part of a rectangular box.Let me think. If I consider the prism as part of a rectangular box, the base would be the right triangle ABC, and the height is 2. So, the dimensions of the box would be AB=3, AC=4, and AA₁=2. Then, the space diagonal of this box would be √(3² + 4² + 2²) = √(9 + 16 + 4) = √29. Therefore, the diameter of the circumscribed sphere is √29, so the radius is half of that, which is √29 / 2.Okay, so the radius R of the circumscribed sphere is √29 / 2. Then, the surface area of a sphere is 4πR², so plugging in, we get 4π*(√29 / 2)² = 4π*(29/4) = 29π. So, the surface area of the circumscribed sphere is 29π.Now, moving on to the inscribed sphere. An inscribed sphere (in-sphere) of a prism touches all the faces of the prism. For a prism, the in-sphere must be tangent to all the lateral faces and the two bases. However, not all prisms have an inscribed sphere; only those that are "tangent" prisms, meaning that the in-sphere touches all faces.Wait, I need to check if this prism can have an inscribed sphere. For a prism to have an inscribed sphere, it must be that the in-sphere touches all the faces, which requires that the prism is both equilateral and equiangular, but in this case, the base is a right triangle, which is not equilateral. Hmm, so maybe it's not possible to have an inscribed sphere? Or perhaps I'm misunderstanding.Wait, actually, in 3D, a prism can have an inscribed sphere if and only if the base is a tangential polygon, meaning it has an incircle, and the height of the prism is equal to twice the radius of the incircle. Wait, no, that might not be exactly right.Let me recall. For a prism to have an inscribed sphere, the base must be a tangential polygon, and the height must be equal to twice the inradius of the base. So, in this case, the base is a right triangle, which is a tangential polygon because all triangles are tangential (they have an incircle). So, the inradius of the base triangle ABC can be found, and then if the height of the prism is equal to twice that inradius, then the prism can have an inscribed sphere.Wait, but in this case, the height of the prism is 2 units. Let me compute the inradius of the base triangle ABC.The inradius r of a triangle is given by the formula r = A / s, where A is the area of the triangle, and s is the semi-perimeter.First, let's compute the area of triangle ABC. Since it's a right triangle with legs 3 and 4, the area A is (3*4)/2 = 6.Next, the semi-perimeter s is (AB + AC + BC)/2 = (3 + 4 + 5)/2 = 12/2 = 6.Therefore, the inradius r = A / s = 6 / 6 = 1.So, the inradius of the base triangle is 1 unit. Therefore, for the prism to have an inscribed sphere, the height of the prism should be equal to twice the inradius, which would be 2*1=2 units. Hey, that's exactly the height of the prism given in the problem! So, yes, this prism does have an inscribed sphere.Therefore, the radius of the inscribed sphere is equal to the inradius of the base triangle, which is 1 unit. Wait, is that correct? Or is it different?Wait, no. In a prism, the inradius of the inscribed sphere is actually equal to the inradius of the base polygon, but only if the height of the prism is twice the inradius. Wait, let me think again.In a prism with a tangential base, if the height h is equal to 2r, where r is the inradius of the base, then the inradius of the prism's inscribed sphere is r. So, in this case, since h = 2r = 2*1=2, which matches the given height, the inradius of the inscribed sphere is indeed 1.Therefore, the radius r of the inscribed sphere is 1. Then, the surface area of the inscribed sphere is 4πr² = 4π*(1)² = 4π.So, the surface area of the inscribed sphere is 4π, and the surface area of the circumscribed sphere is 29π. Therefore, the sum of these two surface areas is 4π + 29π = 33π.Wait, that seems straightforward, but let me double-check to make sure I didn't make any mistakes.First, checking the circumradius. The prism is considered as part of a rectangular box with sides 3, 4, and 2. The space diagonal is √(3² + 4² + 2²) = √(9 + 16 + 4) = √29. So, the radius is √29 / 2, and the surface area is 4π*(√29 / 2)² = 4π*(29/4) = 29π. That seems correct.Next, the inradius. The base triangle has inradius 1, and since the height of the prism is 2, which is twice the inradius, the prism can have an inscribed sphere with radius 1. Therefore, the surface area is 4π*(1)² = 4π. That also seems correct.Adding them together, 29π + 4π = 33π. So, the sum of the surface areas is 33π.I think that's the correct answer. I don't see any mistakes in the reasoning. The key was recognizing that the prism can have an inscribed sphere because its base is a tangential polygon and the height is twice the inradius of the base. Then, using the space diagonal to find the circumradius.**Final Answer**The sum of the surface areas of the inscribed sphere and the circumscribed sphere of the prism is boxed{33pi}.