💡Alright, so I have this geometry problem here. It says that in a right angle ∠POQ, there's a moving point C. I need to find points A on OP and B on OQ such that the sum of the lengths BC and CA equals a constant length l, and the area of the quadrilateral AOB C is maximized. Hmm, okay, let me try to visualize this.First, I imagine a right angle with vertex O, and two arms OP and OQ forming a 90-degree angle. Point C is somewhere inside this angle, and it's moving. I need to choose points A on OP and B on OQ such that the path from B to C to A has a total length of l. At the same time, the area of the quadrilateral formed by points A, O, B, and C should be as large as possible.I think drawing a diagram might help. Let me sketch this out mentally. Points A and B are on OP and OQ respectively. So, quadrilateral AOB C is a four-sided figure with vertices at A, O, B, and C. The sides are AO, OB, BC, and CA. The perimeter condition is BC + CA = l.Wait, actually, the perimeter is only BC + CA, not the entire perimeter of the quadrilateral. So, AO and OB are not part of the perimeter constraint. That’s interesting. So, AO and OB can vary as long as BC + CA is fixed at l.I need to maximize the area of quadrilateral AOB C. The area of a quadrilateral can sometimes be tricky, but maybe I can break it down into triangles or something else. Since AOB is a right triangle (because OP and OQ form a right angle), the area of AOB is (1/2)*OA*OB. Then, the area of triangle BOC and triangle AOC might also come into play. Hmm, not sure yet.Alternatively, maybe I can parametrize the positions of A and B. Let me assign coordinates to make this more concrete. Let’s place point O at the origin (0,0). Then, OP can be along the x-axis, and OQ along the y-axis. So, point A can be at (a, 0) and point B at (0, b), where a and b are positive real numbers.Point C is somewhere inside the first quadrant. Let's denote its coordinates as (x, y). Given that C is moving, x and y can vary, but we have the condition that BC + CA = l.So, BC is the distance from B to C, which is sqrt((x - 0)^2 + (y - b)^2) = sqrt(x^2 + (y - b)^2). Similarly, CA is the distance from C to A, which is sqrt((x - a)^2 + (y - 0)^2) = sqrt((x - a)^2 + y^2). Therefore, the condition is:sqrt(x^2 + (y - b)^2) + sqrt((x - a)^2 + y^2) = lThat's a bit complicated. Maybe there's a way to simplify this or find a relationship between a and b.Also, the area of quadrilateral AOB C. Since AOB is a right triangle, its area is (1/2)*a*b. Then, the area of quadrilateral AOB C would be the area of AOB plus the area of triangle BOC and triangle AOC? Wait, no, because C is inside the angle, so maybe it's just the area of AOB minus something? Hmm, I'm getting confused.Alternatively, maybe I can use the shoelace formula to calculate the area of quadrilateral AOB C. The coordinates of the quadrilateral are O(0,0), A(a,0), B(0,b), and C(x,y). So, applying the shoelace formula:Area = (1/2)| (0*0 + a*b + 0*y + x*0) - (0*a + 0*0 + b*x + y*0) |= (1/2)|0 + a*b + 0 + 0 - 0 - 0 - b*x - 0|= (1/2)|a*b - b*x|= (1/2)b(a - x)Wait, that seems too simple. Is that correct? Let me double-check. The shoelace formula is:Area = (1/2)|sum over i (x_i y_{i+1} - x_{i+1} y_i)|So, plugging in the coordinates:O(0,0), A(a,0), B(0,b), C(x,y), back to O(0,0).Calculating the terms:(0*0 - a*0) + (a*b - 0*0) + (0*y - b*x) + (x*0 - y*0) + (0*0 - 0*y)Simplify each term:0 + (a*b - 0) + (0 - b*x) + 0 + 0 = a*b - b*xSo, the area is (1/2)|a*b - b*x|. Since all points are in the first quadrant, a, b, x, y are positive, so the absolute value can be removed:Area = (1/2)(a*b - b*x) = (1/2)b(a - x)Hmm, okay, so the area depends on a, b, and x. But we have the condition that sqrt(x^2 + (y - b)^2) + sqrt((x - a)^2 + y^2) = l.This seems complicated. Maybe I can find a relationship between a and b to maximize the area.Alternatively, perhaps using reflection to simplify the problem. I remember that in some optimization problems involving distances, reflecting a point can help find the shortest path or something similar.Let me think. If I reflect point C over OP or OQ, maybe that can help. Let's try reflecting C over OP. The reflection of C(x,y) over OP (the x-axis) would be C'(x, -y). Similarly, reflecting over OQ (the y-axis) would be C''(-x, y).But I'm not sure how that helps yet. Alternatively, maybe reflecting point A or B.Wait, another idea: since we have BC + CA = l, which is similar to the definition of an ellipse, where the sum of distances from two foci is constant. So, if I fix points B and A, then point C lies on an ellipse with foci at B and A, and major axis length l.But in this case, points A and B are variable, so it's not straightforward.Alternatively, maybe I can fix point C and then find A and B such that BC + CA = l. But the problem says point C is moving, so I think we need a general solution.Wait, maybe I can parameterize point C. Let me assume that point C is somewhere along the angle bisector of ∠POQ. Since ∠POQ is 90 degrees, the bisector would make 45 degrees with both OP and OQ.So, if point C is on the bisector, its coordinates would be (k, k) for some k > 0.Then, points A and B would be on OP and OQ respectively, so A is (a, 0) and B is (0, b). Then, the distances BC and CA can be expressed in terms of a, b, and k.Let me compute BC first. Point B is (0, b), point C is (k, k). So,BC = sqrt((k - 0)^2 + (k - b)^2) = sqrt(k^2 + (k - b)^2)Similarly, CA is the distance from C(k, k) to A(a, 0):CA = sqrt((a - k)^2 + (0 - k)^2) = sqrt((a - k)^2 + k^2)Given that BC + CA = l, we have:sqrt(k^2 + (k - b)^2) + sqrt((a - k)^2 + k^2) = lThat's still complicated, but maybe I can find a relationship between a and b.Also, the area of quadrilateral AOB C is (1/2)b(a - k), as we found earlier.So, to maximize the area, we need to maximize (1/2)b(a - k), given the constraint on the sum of distances.Hmm, perhaps using calculus here. Let me denote S = (1/2)b(a - k), which we want to maximize.But we have the constraint:sqrt(k^2 + (k - b)^2) + sqrt((a - k)^2 + k^2) = lThis seems like a problem that can be approached with Lagrange multipliers, where we maximize S subject to the constraint.Let me set up the Lagrangian:L = (1/2)b(a - k) + λ [sqrt(k^2 + (k - b)^2) + sqrt((a - k)^2 + k^2) - l]Then, take partial derivatives with respect to a, b, k, and λ, set them equal to zero, and solve.First, partial derivative with respect to a:∂L/∂a = (1/2)b + λ * [ ( (a - k) ) / sqrt((a - k)^2 + k^2) ) ] = 0Similarly, partial derivative with respect to b:∂L/∂b = (1/2)(a - k) + λ * [ ( (k - b) ) / sqrt(k^2 + (k - b)^2) ) ] = 0Partial derivative with respect to k:∂L/∂k = (1/2)(-b) + λ * [ (k + (k - b)) / sqrt(k^2 + (k - b)^2) + ( - (a - k) ) / sqrt((a - k)^2 + k^2) ) ] = 0And partial derivative with respect to λ:∂L/∂λ = sqrt(k^2 + (k - b)^2) + sqrt((a - k)^2 + k^2) - l = 0This is getting really messy. Maybe there's a symmetry or a substitution that can simplify this.Wait, if point C is on the angle bisector, then maybe OA = OB? Because of the symmetry, perhaps the maximum area occurs when OA = OB.Let me assume OA = OB = t. Then, point A is (t, 0), point B is (0, t), and point C is (k, k).Then, the distances BC and CA become:BC = sqrt(k^2 + (k - t)^2)CA = sqrt((t - k)^2 + k^2)So, BC + CA = 2*sqrt((t - k)^2 + k^2) = lSo, sqrt((t - k)^2 + k^2) = l/2Squaring both sides:(t - k)^2 + k^2 = (l/2)^2Expanding:t^2 - 2tk + k^2 + k^2 = l^2 / 4Simplify:t^2 - 2tk + 2k^2 = l^2 / 4Now, the area S is (1/2)*t*(t - k) = (1/2)(t^2 - t*k)We need to express S in terms of t and k, but we have the equation t^2 - 2tk + 2k^2 = l^2 / 4.Let me solve for t in terms of k or vice versa.From t^2 - 2tk + 2k^2 = l^2 / 4, let's treat this as a quadratic in t:t^2 - 2k t + (2k^2 - l^2 / 4) = 0Using quadratic formula:t = [2k ± sqrt(4k^2 - 4*(2k^2 - l^2 / 4))]/2= [2k ± sqrt(4k^2 - 8k^2 + l^2)]/2= [2k ± sqrt(-4k^2 + l^2)]/2Since t must be real, the discriminant must be non-negative:-4k^2 + l^2 ≥ 0 ⇒ l^2 ≥ 4k^2 ⇒ k ≤ l/2So, k can vary from 0 to l/2.Now, t = [2k ± sqrt(l^2 - 4k^2)]/2But since t must be positive, and sqrt(l^2 - 4k^2) is positive, we take the positive sign:t = [2k + sqrt(l^2 - 4k^2)]/2 = k + (sqrt(l^2 - 4k^2))/2Now, plug this into the area S:S = (1/2)(t^2 - t*k)First, compute t^2:t^2 = [k + (sqrt(l^2 - 4k^2))/2]^2= k^2 + k*sqrt(l^2 - 4k^2) + (l^2 - 4k^2)/4= k^2 + k*sqrt(l^2 - 4k^2) + l^2/4 - k^2= k*sqrt(l^2 - 4k^2) + l^2/4Then, compute t*k:t*k = [k + (sqrt(l^2 - 4k^2))/2]*k= k^2 + (k*sqrt(l^2 - 4k^2))/2So, S = (1/2)(t^2 - t*k) = (1/2)[(k*sqrt(l^2 - 4k^2) + l^2/4) - (k^2 + (k*sqrt(l^2 - 4k^2))/2)]Simplify inside the brackets:= (1/2)[k*sqrt(l^2 - 4k^2) + l^2/4 - k^2 - (k*sqrt(l^2 - 4k^2))/2]= (1/2)[(k*sqrt(l^2 - 4k^2) - (k*sqrt(l^2 - 4k^2))/2) + l^2/4 - k^2]= (1/2)[(k*sqrt(l^2 - 4k^2))/2 + l^2/4 - k^2]= (1/2)*(k*sqrt(l^2 - 4k^2))/2 + (1/2)*(l^2/4 - k^2)= (k*sqrt(l^2 - 4k^2))/4 + l^2/8 - k^2/2So, S(k) = (k*sqrt(l^2 - 4k^2))/4 + l^2/8 - k^2/2This is a function of k, and we need to find its maximum for k in [0, l/2].To find the maximum, take the derivative of S with respect to k and set it to zero.First, let me denote f(k) = (k*sqrt(l^2 - 4k^2))/4 + l^2/8 - k^2/2Compute f'(k):f'(k) = [sqrt(l^2 - 4k^2) + k*(1/(2*sqrt(l^2 - 4k^2)))*(-8k)]/4 + 0 - kSimplify:= [sqrt(l^2 - 4k^2) - (4k^2)/sqrt(l^2 - 4k^2)]/4 - kFactor out 1/sqrt(l^2 - 4k^2):= [ (l^2 - 4k^2) - 4k^2 ] / (4*sqrt(l^2 - 4k^2)) - k= [ l^2 - 8k^2 ] / (4*sqrt(l^2 - 4k^2)) - kSet f'(k) = 0:[ l^2 - 8k^2 ] / (4*sqrt(l^2 - 4k^2)) - k = 0Multiply both sides by 4*sqrt(l^2 - 4k^2):l^2 - 8k^2 - 4k*sqrt(l^2 - 4k^2) = 0Let me rearrange:l^2 - 8k^2 = 4k*sqrt(l^2 - 4k^2)Square both sides to eliminate the square root:(l^2 - 8k^2)^2 = 16k^2(l^2 - 4k^2)Expand the left side:= l^4 - 16l^2 k^2 + 64k^4Right side:= 16k^2 l^2 - 64k^4Set them equal:l^4 - 16l^2 k^2 + 64k^4 = 16k^2 l^2 - 64k^4Bring all terms to the left:l^4 - 16l^2 k^2 + 64k^4 - 16k^2 l^2 + 64k^4 = 0Combine like terms:l^4 - 32l^2 k^2 + 128k^4 = 0Divide both sides by l^4 (assuming l ≠ 0):1 - 32(k^2/l^2) + 128(k^4/l^4) = 0Let me set u = k^2/l^2. Then, the equation becomes:1 - 32u + 128u^2 = 0This is a quadratic in u:128u^2 - 32u + 1 = 0Solve for u:u = [32 ± sqrt(1024 - 512)] / (2*128)= [32 ± sqrt(512)] / 256= [32 ± 16*sqrt(2)] / 256= [2 ± sqrt(2)] / 16So, u = (2 + sqrt(2))/16 or u = (2 - sqrt(2))/16Since u = k^2/l^2 must be positive and less than 1/4 (because k ≤ l/2), let's check:(2 + sqrt(2))/16 ≈ (2 + 1.414)/16 ≈ 3.414/16 ≈ 0.213, which is less than 1/4 (0.25). Okay.(2 - sqrt(2))/16 ≈ (2 - 1.414)/16 ≈ 0.586/16 ≈ 0.0366, also less than 1/4.So both solutions are valid. But we need to check which one gives a maximum.Let me compute f''(k) or test the values, but maybe it's easier to plug back into the equation.Wait, when we squared both sides, we might have introduced extraneous solutions, so we need to verify.Let me first take u = (2 + sqrt(2))/16:Then, k = l*sqrt(u) = l*sqrt((2 + sqrt(2))/16) = l*(sqrt(2 + sqrt(2)))/4Similarly, for u = (2 - sqrt(2))/16:k = l*sqrt((2 - sqrt(2))/16) = l*(sqrt(2 - sqrt(2)))/4Now, let's compute t for each case.Recall t = k + (sqrt(l^2 - 4k^2))/2First, for u = (2 + sqrt(2))/16:k = l*(sqrt(2 + sqrt(2)))/4Compute 4k^2:4k^2 = 4*l^2*(2 + sqrt(2))/16 = l^2*(2 + sqrt(2))/4So, sqrt(l^2 - 4k^2) = sqrt(l^2 - l^2*(2 + sqrt(2))/4) = l*sqrt(1 - (2 + sqrt(2))/4) = l*sqrt((4 - 2 - sqrt(2))/4) = l*sqrt((2 - sqrt(2))/4) = l*(sqrt(2 - sqrt(2)))/2Thus, t = k + (sqrt(l^2 - 4k^2))/2 = l*(sqrt(2 + sqrt(2)))/4 + [l*(sqrt(2 - sqrt(2)))/2]/2 = l*(sqrt(2 + sqrt(2)))/4 + l*(sqrt(2 - sqrt(2)))/4 = l*(sqrt(2 + sqrt(2)) + sqrt(2 - sqrt(2)))/4Similarly, for u = (2 - sqrt(2))/16:k = l*(sqrt(2 - sqrt(2)))/4Compute 4k^2:4k^2 = 4*l^2*(2 - sqrt(2))/16 = l^2*(2 - sqrt(2))/4sqrt(l^2 - 4k^2) = sqrt(l^2 - l^2*(2 - sqrt(2))/4) = l*sqrt(1 - (2 - sqrt(2))/4) = l*sqrt((4 - 2 + sqrt(2))/4) = l*sqrt((2 + sqrt(2))/4) = l*(sqrt(2 + sqrt(2)))/2Thus, t = k + (sqrt(l^2 - 4k^2))/2 = l*(sqrt(2 - sqrt(2)))/4 + [l*(sqrt(2 + sqrt(2)))/2]/2 = l*(sqrt(2 - sqrt(2)))/4 + l*(sqrt(2 + sqrt(2)))/4 = l*(sqrt(2 - sqrt(2)) + sqrt(2 + sqrt(2)))/4So, in both cases, t is the same:t = l*(sqrt(2 + sqrt(2)) + sqrt(2 - sqrt(2)))/4Let me compute sqrt(2 + sqrt(2)) + sqrt(2 - sqrt(2)):Let’s denote S = sqrt(2 + sqrt(2)) + sqrt(2 - sqrt(2))Compute S^2:= (sqrt(2 + sqrt(2)) + sqrt(2 - sqrt(2)))^2= (2 + sqrt(2)) + (2 - sqrt(2)) + 2*sqrt{(2 + sqrt(2))(2 - sqrt(2))}= 4 + 2*sqrt{4 - 2}= 4 + 2*sqrt{2}Thus, S = sqrt(4 + 2*sqrt(2)) = sqrt(2*(2 + sqrt(2))) = sqrt(2)*sqrt(2 + sqrt(2))Wait, but that seems like a loop. Alternatively, maybe we can express S in terms of known quantities.Alternatively, note that sqrt(2 + sqrt(2)) = 2*cos(22.5°) and sqrt(2 - sqrt(2)) = 2*sin(22.5°). Because:cos(22.5°) = sqrt(2 + sqrt(2))/2sin(22.5°) = sqrt(2 - sqrt(2))/2So, sqrt(2 + sqrt(2)) = 2*cos(22.5°)sqrt(2 - sqrt(2)) = 2*sin(22.5°)Thus, S = 2*cos(22.5°) + 2*sin(22.5°) = 2*(cos(22.5°) + sin(22.5°))We can compute cos(22.5°) + sin(22.5°):= sqrt(2)*sin(45° + 22.5°) = sqrt(2)*sin(67.5°)But sin(67.5°) = sin(45° + 22.5°) = sin(45°)cos(22.5°) + cos(45°)sin(22.5°) = (sqrt(2)/2)(cos(22.5°) + sin(22.5°))Wait, that might not help. Alternatively, compute numerically:cos(22.5°) ≈ 0.924, sin(22.5°) ≈ 0.383, so sum ≈ 1.307Thus, S ≈ 2*1.307 ≈ 2.614But exact value is 2*(cos(22.5°) + sin(22.5°)) = 2*sqrt(2)*sin(67.5°) as above.But maybe we can leave it as is.So, t = l*(sqrt(2 + sqrt(2)) + sqrt(2 - sqrt(2)))/4 = l*(2*cos(22.5°) + 2*sin(22.5°))/4 = l*(cos(22.5°) + sin(22.5°))/2Alternatively, t = l*(sqrt(2 + sqrt(2)) + sqrt(2 - sqrt(2)))/4Either way, it's a constant multiple of l.Now, going back, we have two possible values for k: k = l*(sqrt(2 + sqrt(2)))/4 and k = l*(sqrt(2 - sqrt(2)))/4But we need to check which one gives a maximum.Let me compute the second derivative or test the values.Alternatively, since we have two critical points, we can compute S(k) for both and see which is larger.Compute S(k) for u = (2 + sqrt(2))/16:k = l*(sqrt(2 + sqrt(2)))/4Compute S(k):S = (k*sqrt(l^2 - 4k^2))/4 + l^2/8 - k^2/2We already computed sqrt(l^2 - 4k^2) = l*(sqrt(2 - sqrt(2)))/2So,S = [l*(sqrt(2 + sqrt(2)))/4 * l*(sqrt(2 - sqrt(2)))/2 ] /4 + l^2/8 - [l^2*(2 + sqrt(2))/16]/2Simplify:First term:= [l^2 * sqrt{(2 + sqrt(2))(2 - sqrt(2))} ] / (4*2*4)= [l^2 * sqrt{4 - 2} ] / 32= [l^2 * sqrt{2} ] / 32Second term: l^2/8Third term:= [l^2*(2 + sqrt(2))/16]/2= l^2*(2 + sqrt(2))/32So, S = (l^2*sqrt(2))/32 + l^2/8 - l^2*(2 + sqrt(2))/32Combine terms:= l^2/8 + [ (sqrt(2) - 2 - sqrt(2)) ] /32 * l^2= l^2/8 - (2)/32 * l^2= l^2/8 - l^2/16= l^2/16Similarly, for u = (2 - sqrt(2))/16:k = l*(sqrt(2 - sqrt(2)))/4Compute S(k):sqrt(l^2 - 4k^2) = l*(sqrt(2 + sqrt(2)))/2So,S = [l*(sqrt(2 - sqrt(2)))/4 * l*(sqrt(2 + sqrt(2)))/2 ] /4 + l^2/8 - [l^2*(2 - sqrt(2))/16]/2Simplify:First term:= [l^2 * sqrt{(2 - sqrt(2))(2 + sqrt(2))} ] / (4*2*4)= [l^2 * sqrt{4 - 2} ] /32= [l^2 * sqrt{2} ] /32Second term: l^2/8Third term:= [l^2*(2 - sqrt(2))/16]/2= l^2*(2 - sqrt(2))/32So, S = (l^2*sqrt(2))/32 + l^2/8 - l^2*(2 - sqrt(2))/32Combine terms:= l^2/8 + [ sqrt(2) - 2 + sqrt(2) ] /32 * l^2= l^2/8 + [ 2*sqrt(2) - 2 ] /32 * l^2= l^2/8 + (sqrt(2) - 1)/16 * l^2Convert l^2/8 to 2*l^2/16:= 2*l^2/16 + (sqrt(2) - 1)*l^2/16= [2 + sqrt(2) - 1]*l^2/16= [1 + sqrt(2)]*l^2/16Which is approximately (1 + 1.414)/16 ≈ 2.414/16 ≈ 0.1509*l^2Comparing to the previous case where S = l^2/16 ≈ 0.0625*l^2, clearly the second case gives a larger area.Therefore, the maximum area occurs when u = (2 - sqrt(2))/16, i.e., k = l*(sqrt(2 - sqrt(2)))/4Thus, the maximum area is [1 + sqrt(2)]*l^2/16But wait, let me double-check the calculations because I might have made a mistake in combining terms.Wait, in the first case, S = l^2/16 ≈ 0.0625*l^2In the second case, S = [1 + sqrt(2)]*l^2/16 ≈ (1 + 1.414)/16 ≈ 2.414/16 ≈ 0.1509*l^2Yes, that's correct. So, the maximum area is [1 + sqrt(2)]*l^2/16But let me express it differently:[1 + sqrt(2)]/16 = (sqrt(2) + 1)/16Alternatively, factor out 1/16:= (sqrt(2) + 1)/16 * l^2But I think it's better to rationalize or express it in terms of trigonometric functions.Wait, earlier we saw that t = l*(sqrt(2 + sqrt(2)) + sqrt(2 - sqrt(2)))/4Which is equal to l*(2*cos(22.5°) + 2*sin(22.5°))/4 = l*(cos(22.5°) + sin(22.5°))/2And since cos(22.5°) + sin(22.5°) = sqrt(2)*sin(67.5°), as I thought earlier.But maybe it's better to leave it as is.So, in conclusion, the maximum area is achieved when OA = OB = t = l*(sqrt(2 + sqrt(2)) + sqrt(2 - sqrt(2)))/4, and point C is on the angle bisector at (k, k) where k = l*(sqrt(2 - sqrt(2)))/4But let me see if I can express t in a simpler form.We have:t = l*(sqrt(2 + sqrt(2)) + sqrt(2 - sqrt(2)))/4Let me compute sqrt(2 + sqrt(2)) + sqrt(2 - sqrt(2)):Let’s denote S = sqrt(2 + sqrt(2)) + sqrt(2 - sqrt(2))As before, S^2 = 4 + 2*sqrt(2)So, S = sqrt(4 + 2*sqrt(2)) = sqrt(2*(2 + sqrt(2))) = sqrt(2)*sqrt(2 + sqrt(2))But that doesn't seem to simplify further.Alternatively, note that sqrt(4 + 2*sqrt(2)) = 2*cos(22.5°) + 2*sin(22.5°), but that might not help.Alternatively, express t in terms of l and trigonometric functions:t = l*(sqrt(2 + sqrt(2)) + sqrt(2 - sqrt(2)))/4 = l*(2*cos(22.5°) + 2*sin(22.5°))/4 = l*(cos(22.5°) + sin(22.5°))/2Which is a neat expression.So, OA = OB = l*(cos(22.5°) + sin(22.5°))/2And point C is at (k, k) where k = l*(sqrt(2 - sqrt(2)))/4Alternatively, since sqrt(2 - sqrt(2)) = 2*sin(22.5°), as we saw earlier, so k = l*(2*sin(22.5°))/4 = l*sin(22.5°)/2Similarly, sqrt(2 + sqrt(2)) = 2*cos(22.5°), so t = l*(2*cos(22.5°) + 2*sin(22.5°))/4 = l*(cos(22.5°) + sin(22.5°))/2So, in terms of trigonometric functions, OA = OB = l*(cos(22.5°) + sin(22.5°))/2 and OC = k = l*sin(22.5°)/2But wait, OC is the distance from O to C, which is sqrt(k^2 + k^2) = k*sqrt(2) = l*sin(22.5°)/2 * sqrt(2) = l*sin(22.5°)*sqrt(2)/2But sin(22.5°)*sqrt(2) = sin(45°) = sqrt(2)/2, because sin(22.5°)*sqrt(2) = 2*sin(22.5°)*cos(22.5°) = sin(45°) = sqrt(2)/2Wait, let me verify:sin(2θ) = 2*sinθ*cosθSo, sin(45°) = 2*sin(22.5°)*cos(22.5°)Thus, sin(22.5°)*cos(22.5°) = sin(45°)/2 = (sqrt(2)/2)/2 = sqrt(2)/4But in our case, OC = l*sin(22.5°)*sqrt(2)/2= l*(sqrt(2)/2)*sin(22.5°)= l*(sqrt(2)/2)*(sqrt(2 - sqrt(2))/2)= l*(sqrt(2)*(sqrt(2 - sqrt(2))))/4= l*sqrt(2*(2 - sqrt(2)))/4= l*sqrt(4 - 2*sqrt(2))/4But sqrt(4 - 2*sqrt(2)) = sqrt(2)*sqrt(2 - sqrt(2)) which doesn't seem to help.Alternatively, maybe it's better to leave OC as l*sin(22.5°)/2*sqrt(2)But I think the key takeaway is that OA = OB = l*(cos(22.5°) + sin(22.5°))/2 and point C is on the angle bisector at a distance OC = l*sin(45°)/2, but I need to verify.Wait, earlier we had OC = k*sqrt(2) = l*sin(22.5°)/2 * sqrt(2) = l*(sqrt(2)/2)*sin(22.5°)But sin(22.5°) = sqrt(2 - sqrt(2))/2, so:OC = l*(sqrt(2)/2)*(sqrt(2 - sqrt(2))/2) = l*(sqrt(2)*sqrt(2 - sqrt(2)))/4= l*sqrt(2*(2 - sqrt(2)))/4= l*sqrt(4 - 2*sqrt(2))/4Which is the same as before.Alternatively, note that 4 - 2*sqrt(2) = (sqrt(2))^2 - 2*sqrt(2) + 1 -1 = (sqrt(2) -1)^2 -1, but that might not help.In any case, the exact expression for OC is l*sqrt(4 - 2*sqrt(2))/4But perhaps it's better to express it in terms of trigonometric functions.In summary, the maximum area is achieved when OA = OB = l*(cos(22.5°) + sin(22.5°))/2 and point C is on the angle bisector at a distance OC = l*sqrt(4 - 2*sqrt(2))/4 from O.But to express OA and OB in a simplified form, let's compute cos(22.5°) + sin(22.5°):cos(22.5°) + sin(22.5°) = sqrt(2)*sin(67.5°) as we saw earlierBut sin(67.5°) = sin(45° + 22.5°) = sin(45°)cos(22.5°) + cos(45°)sin(22.5°) = (sqrt(2)/2)(cos(22.5°) + sin(22.5°))Wait, that's circular.Alternatively, compute numerically:cos(22.5°) ≈ 0.924, sin(22.5°) ≈ 0.383, so sum ≈ 1.307Thus, OA = OB ≈ l*1.307/2 ≈ l*0.6535But to express it exactly, we can write:OA = OB = l*(sqrt(2 + sqrt(2)) + sqrt(2 - sqrt(2)))/4Which is the exact value.So, in conclusion, the points A and B should be chosen such that OA = OB = l*(sqrt(2 + sqrt(2)) + sqrt(2 - sqrt(2)))/4, and point C lies on the angle bisector at a distance OC = l*sqrt(4 - 2*sqrt(2))/4 from O.This configuration maximizes the area of quadrilateral AOB C under the given perimeter constraint BC + CA = l.**Final Answer**The points ( A ) and ( B ) should be chosen such that ( OA = OB = frac{l}{4 sin 22.5^circ} ). Thus, the maximum area is achieved when ( OA = OB = boxed{frac{l}{4 sin 22.5^circ}} ).

💡First, calculate the weekly cost of milk:1/2 gallon * 4/gallon = 2.Subtract the milk cost from the total weekly coffee expense to find the cost of coffee beans:18 - 2 = 16.Determine the number of coffee bags purchased:16 / 8 per bag = 2 bags.Calculate the total ounces of coffee beans:2 bags * 10.5 ounces per bag = 21 ounces.Find the total number of cups of coffee made per week:2 cups/day * 7 days = 14 cups.Finally, determine the ounces of coffee beans per cup:21 ounces / 14 cups = 1.5 ounces per cup.

💡First, let's define ( x = sintheta + costheta ).Next, we square both sides of the equation:[x^2 = (sintheta + costheta)^2 = sin^2theta + 2sinthetacostheta + cos^2theta]Using the Pythagorean identity ( sin^2theta + cos^2theta = 1 ), we can simplify the equation:[x^2 = 1 + 2sinthetacostheta]We know that ( sinthetacostheta ) can be expressed using the double-angle identity for cosine:[sinthetacostheta = frac{1 - cos 2theta}{2}]Substituting this back into our equation for ( x^2 ):[x^2 = 1 + 2 left( frac{1 - cos 2theta}{2} right) = 1 + 1 - cos 2theta = 2 - cos 2theta]Taking the square root of both sides to solve for ( x ):[x = sqrt{2 - cos 2theta}]Since ( theta ) is an acute angle, both ( sintheta ) and ( costheta ) are non-negative, so:[sintheta + costheta = sqrt{2 - cos 2theta}]Given that ( cos 2theta = b ), we substitute ( b ) into the equation:[sintheta + costheta = sqrt{2 - b}]

💡First, I need to determine the total number of animals Michael has. He has 2 cats and an unknown number of dogs, which I'll call ( d ).The total cost for pet-sitting is 65, and the charge is 13 per animal per night. So, the equation representing the total cost is:[2 times 13 + d times 13 = 65]Simplifying the equation:[26 + 13d = 65]Next, I'll solve for ( d ) by subtracting 26 from both sides:[13d = 39]Finally, I'll divide both sides by 13 to find the number of dogs:[d = 3]Therefore, Michael has 3 dogs.

💡Okay, so I need to compute (1 + i)^6. Hmm, I remember that when dealing with complex numbers, especially powers of complex numbers, there are a few methods I can use. One common approach is to use the binomial theorem, but that might get a bit messy for the sixth power. Another method is to convert the complex number into polar form and then use De Moivre's theorem, which might be more straightforward for exponentiation. Let me think about both methods and see which one is easier.First, let me recall what (1 + i) looks like in polar form. To convert a complex number a + bi to polar form, I need to find its magnitude and its angle. The magnitude r is given by sqrt(a^2 + b^2), and the angle θ is arctangent(b/a). For (1 + i), both a and b are 1, so the magnitude r is sqrt(1^2 + 1^2) = sqrt(2). The angle θ is arctangent(1/1) which is 45 degrees or π/4 radians. So, in polar form, (1 + i) is sqrt(2) * (cos(π/4) + i sin(π/4)).Now, using De Moivre's theorem, which states that (r (cos θ + i sin θ))^n = r^n (cos(nθ) + i sin(nθ)). Applying this to (1 + i)^6, we get [sqrt(2)]^6 * (cos(6 * π/4) + i sin(6 * π/4)).Let me compute [sqrt(2)]^6 first. Since sqrt(2) is 2^(1/2), raising it to the 6th power gives (2^(1/2))^6 = 2^(6/2) = 2^3 = 8. So, the magnitude part is 8.Next, let's compute the angle. 6 * π/4 is equal to (6/4)π, which simplifies to (3/2)π. So, the angle is 3π/2 radians, which is 270 degrees. Now, I need to find cos(3π/2) and sin(3π/2). I remember that cos(3π/2) is 0 and sin(3π/2) is -1. Therefore, the expression becomes 8 * (0 + i*(-1)) = 8*(-i) = -8i.Wait, let me verify this result using another method to make sure I didn't make a mistake. Maybe I can compute (1 + i)^6 step by step by expanding the powers.Starting with (1 + i)^2. Using the binomial expansion, (1 + i)^2 = 1^2 + 2*1*i + i^2 = 1 + 2i + (-1) = 2i. Okay, so (1 + i)^2 is 2i.Then, (1 + i)^4 would be [(1 + i)^2]^2 = (2i)^2 = 4i^2 = 4*(-1) = -4. So, (1 + i)^4 is -4.Now, (1 + i)^6 is (1 + i)^4 * (1 + i)^2 = (-4)*(2i) = -8i. That matches the result I got using polar form and De Moivre's theorem. So, both methods give me the same answer, which is reassuring.Just to be thorough, let me try another approach. Maybe using the binomial theorem directly on (1 + i)^6. The binomial theorem states that (a + b)^n = sum from k=0 to n of (n choose k) * a^(n-k) * b^k. So, for (1 + i)^6, a = 1, b = i, and n = 6.Calculating each term:- When k=0: (6 choose 0)*1^6*i^0 = 1*1*1 = 1- When k=1: (6 choose 1)*1^5*i^1 = 6*1*i = 6i- When k=2: (6 choose 2)*1^4*i^2 = 15*1*(-1) = -15- When k=3: (6 choose 3)*1^3*i^3 = 20*1*(-i) = -20i- When k=4: (6 choose 4)*1^2*i^4 = 15*1*1 = 15- When k=5: (6 choose 5)*1^1*i^5 = 6*1*i = 6i- When k=6: (6 choose 6)*1^0*i^6 = 1*1*(-1) = -1Now, adding all these terms together:1 + 6i - 15 - 20i + 15 + 6i - 1Let's combine like terms:Real parts: 1 - 15 + 15 - 1 = (1 - 1) + (-15 + 15) = 0 + 0 = 0Imaginary parts: 6i - 20i + 6i = (6 + 6 - 20)i = (-8)iSo, the result is 0 - 8i, which is -8i. Again, the same answer. This consistency across different methods makes me confident that the result is correct.I think I've thoroughly checked it using three different methods: polar form with De Moivre's theorem, step-by-step exponentiation, and the binomial theorem. All approaches lead to the same conclusion. Therefore, I can be certain that (1 + i)^6 equals -8i.**Final Answer**The value of (1+i)^6 is boxed{-8i}.

💡Okay, so I have this problem about students studying German and Russian at a university. There are 2500 students in total, and each student studies either German or Russian, or maybe both. I need to find the smallest and largest number of students who could be studying both languages, and then find the difference between those two numbers.Let me try to break this down step by step. First, the total number of students is 2500. Each student studies at least one of the two languages, German or Russian, and some might study both. So, this sounds like a problem where I can use the principle of inclusion-exclusion. I remember that principle from set theory, where the number of elements in the union of two sets is equal to the sum of the elements in each set minus the number of elements in their intersection. So, if I let G be the number of students studying German and R be the number studying Russian, then the total number of students is G + R - both. That is, G + R - (both) = 2500. So, the number of students studying both languages is G + R - 2500. But wait, the problem says that the number of students studying German is between 75% and 80% of the university population, and the number studying Russian is between 35% and 45%. So, I need to figure out the ranges for G and R.Let me calculate these percentages. 75% of 2500 is 0.75 * 2500 = 1875. Similarly, 80% is 0.80 * 2500 = 2000. So, G is between 1875 and 2000. For Russian, 35% is 0.35 * 2500 = 875, and 45% is 0.45 * 2500 = 1125. So, R is between 875 and 1125.Now, I need to find the smallest and largest possible number of students studying both languages. Let me denote the number of students studying both as B. So, from the inclusion-exclusion principle, B = G + R - 2500. To find the smallest B, I think I need to minimize B. Since B = G + R - 2500, to minimize B, I need to minimize G + R. Wait, no, actually, since B is G + R - 2500, to minimize B, I need to minimize G + R. But wait, if G and R are as small as possible, then G + R would be as small as possible, making B as small as possible. Hmm, but wait, let me think again.Wait, if G and R are as large as possible, then G + R would be as large as possible, so B would be as large as possible. Conversely, if G and R are as small as possible, G + R would be as small as possible, making B as small as possible. But wait, no, because B is G + R - 2500. So, if G + R is larger, B is larger, and if G + R is smaller, B is smaller. So, to find the minimum B, I need to take the smallest possible G and R, right? Wait, but that might not be correct because G and R can vary independently. Hmm, maybe I need to think differently.Wait, actually, to minimize B, which is G + R - 2500, I need to minimize G + R. So, to minimize G + R, I should take the smallest possible G and the smallest possible R. So, G_min = 1875 and R_min = 875. So, G + R_min = 1875 + 875 = 2750. Then, B_min = 2750 - 2500 = 250. But wait, that can't be right because if G and R are both at their minimums, the overlap would be 250. But let me check if that's possible.Wait, but if G is 1875 and R is 875, then the maximum possible overlap is the smaller of G and R, which is 875. But 250 is less than 875, so that's possible. So, B_min would be 250.Wait, but I'm getting confused here. Let me think again. The formula is B = G + R - 2500. So, if I take G as 1875 and R as 875, then B = 1875 + 875 - 2500 = 250. So, that's correct. So, the minimum number of students studying both languages is 250.Now, for the maximum number of students studying both languages, B_max, I need to maximize B. Since B = G + R - 2500, to maximize B, I need to maximize G + R. So, I should take the maximum possible G and R. G_max is 2000 and R_max is 1125. So, G + R_max = 2000 + 1125 = 3125. Then, B_max = 3125 - 2500 = 625.Wait, but that seems too low. Because if G is 2000 and R is 1125, the maximum overlap can't exceed the smaller of G and R, which is 1125. But according to the formula, it's 625. Hmm, that seems contradictory. Wait, maybe I'm misunderstanding something.Wait, no, the formula is correct. Let me think about it. If G is 2000 and R is 1125, then the maximum possible overlap is when as many students as possible are in both. So, the maximum overlap is the smaller of G and R, which is 1125. But according to the formula, B = G + R - 2500 = 2000 + 1125 - 2500 = 625. So, that suggests that the maximum overlap is 625, not 1125. That seems contradictory.Wait, maybe I'm making a mistake here. Let me think again. The formula gives B = G + R - 2500. So, if G and R are both at their maximums, then B is 625. But if I try to set B to 1125, then G + R would have to be 2500 + 1125 = 3625. But G can only be up to 2000, and R up to 1125, so G + R can only be up to 3125, which is less than 3625. So, that's not possible. Therefore, the maximum B is indeed 625.Wait, but that doesn't make sense because if G is 2000 and R is 1125, the maximum overlap should be 1125, right? Because all Russian students could be studying German as well. So, why does the formula give 625?Wait, maybe I'm misunderstanding the formula. Let me think again. The formula is B = G + R - 2500. So, if G is 2000 and R is 1125, then B = 2000 + 1125 - 2500 = 625. So, that's correct. But if all 1125 Russian students are also studying German, then the number of students studying German would be 2000, which includes 1125 who are also studying Russian. So, the number of students studying only German would be 2000 - 1125 = 875. Then, the number of students studying only Russian would be 1125 - 1125 = 0. So, total students would be 875 (only German) + 0 (only Russian) + 1125 (both) = 2000. But wait, the total number of students is 2500, so that's not possible. So, that suggests that not all Russian students can be studying German because that would only account for 2000 students, leaving 500 students unaccounted for. So, those 500 students must be studying only Russian. Therefore, the maximum overlap can't be 1125 because that would leave only 2000 students, but we have 2500. So, the maximum overlap is actually 2000 + 1125 - 2500 = 625. So, that makes sense now.Therefore, the maximum number of students studying both languages is 625, and the minimum is 250. So, the difference M' - m' is 625 - 250 = 375.Wait, but in the initial problem statement, the user wrote that the answer was 625. But according to my calculations, it's 375. So, I must have made a mistake somewhere.Wait, let me check my calculations again. G ranges from 1875 to 2000, and R ranges from 875 to 1125. So, for B_min, I took G_min and R_min, which gave me B = 1875 + 875 - 2500 = 250. For B_max, I took G_max and R_max, which gave me B = 2000 + 1125 - 2500 = 625. So, the difference is 625 - 250 = 375.But the user's initial answer was 625. So, maybe I'm misunderstanding the problem. Let me read the problem again."Each of the 2500 students at a university studies either German or Russian, and some study both. The number who study German is between 75 percent and 80 percent of the university population, and the number who study Russian is between 35 percent and 45 percent. Let m' be the smallest number of students who could study both languages, and let M' be the largest number of students who could study both languages. Find M' - m'."Wait, so according to the problem, m' is the smallest number of students who could study both, and M' is the largest. So, I think my approach is correct. So, why does the user's initial answer say 625? Maybe I'm missing something.Wait, perhaps I made a mistake in calculating the ranges for G and R. Let me check that again. 75% of 2500 is 1875, and 80% is 2000. So, G is between 1875 and 2000. 35% of 2500 is 875, and 45% is 1125. So, R is between 875 and 1125. So, that's correct.Wait, but maybe I should consider that the number of students studying German is at least 75%, which is 1875, and at most 80%, which is 2000. Similarly, for Russian, it's at least 35%, which is 875, and at most 45%, which is 1125. So, that's correct.Wait, but perhaps the user's initial answer was wrong. Because according to my calculations, the difference is 375. But let me think again.Wait, if I take G_min = 1875 and R_min = 875, then B_min = 1875 + 875 - 2500 = 250. If I take G_max = 2000 and R_max = 1125, then B_max = 2000 + 1125 - 2500 = 625. So, the difference is 625 - 250 = 375.But wait, maybe I'm supposed to find the maximum possible overlap when G and R are at their extremes. Let me think about it differently. The maximum number of students studying both languages occurs when the overlap is as large as possible. That happens when the smaller set is entirely contained within the larger set. So, if R is 1125, which is larger than G_min of 1875? Wait, no, 1125 is less than 1875. So, if R is 1125, the maximum overlap is 1125, but that would require G to be at least 1125. But G can be as low as 1875, which is more than 1125, so the maximum overlap is 1125. But that would require that all Russian students are also German students. But then, the total number of students would be G + R - B = 1875 + 1125 - 1125 = 1875, which is less than 2500. So, that's not possible because we have 2500 students. Therefore, the maximum overlap can't be 1125 because that would leave only 1875 students, but we have 2500. So, the maximum overlap is when G is as large as possible and R is as large as possible, which gives B_max = 625.Similarly, the minimum overlap occurs when G and R are as small as possible, giving B_min = 250. So, the difference is 625 - 250 = 375.Wait, but I'm confused because the user's initial answer was 625. Maybe I'm missing something. Let me check the initial problem again.Wait, the user wrote: "Each of the 2500 students at a university studies either German or Russian, and some study both. The number who study German is between 75 percent and 80 percent of the university population, and the number who study Russian is between 35 percent and 45 percent. Let m' be the smallest number of students who could study both languages, and let M' be the largest number of students who could study both languages. Find M' - m'."So, according to the problem, the answer is M' - m', which is 625 - 250 = 375. But the user's initial answer was 625. So, perhaps the user made a mistake in their initial answer. Alternatively, maybe I'm misunderstanding the problem.Wait, perhaps the user considered m' as the maximum and M' as the minimum, but that would be inconsistent with the problem statement. The problem says m' is the smallest number and M' is the largest number. So, I think my approach is correct.Wait, another way to think about it is to consider the possible ranges for B. The number of students studying both languages must satisfy B ≥ G + R - 2500. So, the minimum B is when G and R are as small as possible, giving B_min = 1875 + 875 - 2500 = 250. The maximum B is when G and R are as large as possible, giving B_max = 2000 + 1125 - 2500 = 625. So, the difference is 625 - 250 = 375.Therefore, I think the correct answer is 375, not 625. So, perhaps the user's initial answer was incorrect.Wait, but let me think again. Maybe I should consider that the number of students studying German is between 75% and 80%, which is 1875 to 2000, and Russian is between 35% and 45%, which is 875 to 1125. So, to find the minimum overlap, I need to maximize the number of students studying only German and only Russian. That is, to minimize the overlap, I need to have as few students as possible studying both. So, that would occur when G and R are as large as possible, but that seems contradictory.Wait, no, actually, to minimize the overlap, I need to have G and R as small as possible, but that would actually increase the overlap because G + R would be smaller, so B = G + R - 2500 would be smaller. Wait, no, if G and R are smaller, G + R is smaller, so B = G + R - 2500 is smaller. So, that would give a smaller B, which is the minimum overlap.Wait, but if G and R are larger, then G + R is larger, so B = G + R - 2500 is larger, which is the maximum overlap.Wait, so to minimize B, I need to take G and R as small as possible, giving B_min = 250. To maximize B, I take G and R as large as possible, giving B_max = 625. So, the difference is 625 - 250 = 375.Therefore, I think the correct answer is 375. So, the user's initial answer of 625 must have been incorrect.Wait, but let me check with another approach. Let me consider the possible ranges for B.The number of students studying both languages, B, must satisfy:B ≥ G + R - 2500.But also, B cannot exceed the smaller of G and R. So, B ≤ min(G, R).So, let's see. For B_min, we have B ≥ G + R - 2500. To minimize B, we need to minimize G + R. So, G_min = 1875, R_min = 875, so B_min = 1875 + 875 - 2500 = 250. But also, B cannot be less than 0, but 250 is positive, so that's fine.For B_max, we have B ≤ min(G, R). So, if G is 2000 and R is 1125, then min(G, R) = 1125. But from the inclusion-exclusion formula, B = G + R - 2500 = 2000 + 1125 - 2500 = 625. So, B_max is 625, which is less than min(G, R) = 1125. So, that's consistent.Therefore, the maximum overlap is 625, and the minimum is 250, so the difference is 375.Therefore, I think the correct answer is 375, not 625. So, the user's initial answer was incorrect.