💡Okay, so I have this problem about polynomials of degree at most 3. Each coefficient is from the set {0, 1, 2, ..., 12}. I need to find how many such polynomials satisfy P(-1) = -12. Hmm, let me think about how to approach this.First, let me write down what a general polynomial of degree at most 3 looks like. It should be something like P(x) = ax³ + bx² + cx + d, where a, b, c, d are coefficients from 0 to 12. So, each of a, b, c, d can be any integer from 0 up to 12, inclusive.Now, the condition given is P(-1) = -12. Let me substitute x = -1 into the polynomial:P(-1) = a(-1)³ + b(-1)² + c(-1) + d.Calculating each term:- a(-1)³ = -a- b(-1)² = b- c(-1) = -c- d remains dSo, putting it all together:P(-1) = -a + b - c + d.And this is equal to -12. So, the equation we have is:-a + b - c + d = -12.Hmm, maybe I can rearrange this equation to make it easier to handle. Let me bring all the terms to one side:b + d - a - c = -12.Alternatively, I can write it as:b + d = a + c - 12.But I'm not sure if that helps directly. Maybe I can think of it in terms of variables. Let me consider the variables a, b, c, d, each ranging from 0 to 12. So, each coefficient is non-negative and at most 12.I need to find the number of quadruples (a, b, c, d) such that b + d - a - c = -12.Hmm, maybe I can rearrange this equation:b + d = a + c - 12.But since b and d are non-negative, the right-hand side must also be non-negative. So, a + c - 12 ≥ 0, which implies that a + c ≥ 12.So, a and c must add up to at least 12. That's an important constraint.Let me think about how to model this. Maybe I can introduce some substitution to make the equation more manageable. Let me define new variables:Let s = a - 12 and t = c - 12.Wait, but a and c are at least 0, so s and t would be at least -12. Hmm, maybe that's not the best substitution. Alternatively, perhaps I can let s = a + c and t = b + d.Then, from the equation b + d = s - 12, so t = s - 12.But s = a + c, and t = b + d.Since a and c are each between 0 and 12, s can range from 0 to 24. Similarly, t can range from 0 to 24.But from t = s - 12, we have that s must be at least 12, so s ranges from 12 to 24, and t ranges from 0 to 12.So, for each s from 12 to 24, t is determined as s - 12.Therefore, the number of solutions is the sum over s from 12 to 24 of the number of ways to write s as a + c times the number of ways to write t = s - 12 as b + d.So, for each s, the number of (a, c) pairs such that a + c = s is equal to the number of non-negative integer solutions to a + c = s with a ≤ 12 and c ≤ 12.Similarly, the number of (b, d) pairs such that b + d = t = s - 12 is the number of non-negative integer solutions to b + d = t with b ≤ 12 and d ≤ 12.So, let me denote:For a given s, the number of (a, c) pairs is:If s ≤ 12, it's s + 1, but since s starts at 12, for s = 12, it's 13 (from a=0,c=12 up to a=12,c=0). Wait, actually, for s from 0 to 12, the number of solutions is s + 1, and for s from 13 to 24, it's 25 - s.Similarly, for t = s - 12, since s ranges from 12 to 24, t ranges from 0 to 12. So, for each t, the number of (b, d) pairs is t + 1.Therefore, the total number of solutions is the sum from s = 12 to 24 of [number of (a, c) pairs for s] multiplied by [number of (b, d) pairs for t = s - 12].So, let's write this as:Total = Σ [from s=12 to 24] (25 - s) * (s - 12 + 1).Simplify the expression inside the sum:(25 - s) * (s - 11).So, Total = Σ [from s=12 to 24] (25 - s)(s - 11).Let me make a substitution to make the summation easier. Let k = s - 12. Then, when s=12, k=0; when s=24, k=12. So, the sum becomes:Total = Σ [from k=0 to 12] (25 - (12 + k))( (12 + k) - 11 )Simplify inside:25 - 12 - k = 13 - k(12 + k) - 11 = 1 + kSo, Total = Σ [from k=0 to 12] (13 - k)(1 + k)Multiply out the terms:(13 - k)(1 + k) = 13(1 + k) - k(1 + k) = 13 + 13k - k - k² = 13 + 12k - k²Therefore, Total = Σ [from k=0 to 12] (13 + 12k - k²)Now, let's compute this sum term by term:First, sum of 13 from k=0 to 12: 13 * 13 = 169Second, sum of 12k from k=0 to 12: 12 * Σk=0^12 k = 12 * (12*13)/2 = 12 * 78 = 936Third, sum of -k² from k=0 to 12: - Σk=0^12 k² = - (12*13*25)/6 = - (3900)/6 = -650Wait, let me verify the formula for the sum of squares: Σk=1^n k² = n(n+1)(2n+1)/6. So, for n=12:Σk=1^12 k² = 12*13*25/6 = (12/6)*13*25 = 2*13*25 = 650. But since we're summing from k=0, it's the same as from k=1, so Σk=0^12 k² = 650.Therefore, the third term is -650.So, putting it all together:Total = 169 + 936 - 650 = (169 + 936) - 650 = 1105 - 650 = 455.So, the total number of such polynomials is 455.Wait, let me double-check my steps to make sure I didn't make a mistake.1. I started by expressing P(-1) = -12, leading to the equation b + d - a - c = -12.2. Then, I rearranged it to b + d = a + c - 12, noting that a + c must be at least 12.3. I introduced s = a + c and t = b + d, so t = s - 12.4. For each s from 12 to 24, I calculated the number of (a, c) pairs as 25 - s and the number of (b, d) pairs as t + 1 = (s - 12) + 1 = s - 11.5. Then, I set up the summation over s from 12 to 24 of (25 - s)(s - 11).6. I changed variables to k = s - 12, leading to the sum from k=0 to 12 of (13 - k)(1 + k).7. Expanded this to 13 + 12k - k² and summed each term separately.8. Calculated the sums: 13*13=169, 12*(Σk)=12*78=936, and -Σk²=-650.9. Added them up: 169 + 936 = 1105; 1105 - 650 = 455.Seems correct. Alternatively, I can think of it as a stars and bars problem.Wait, another approach: Let me consider the equation b + d - a - c = -12.Let me rearrange it as b + d + 12 = a + c.So, a + c = b + d + 12.Since a, c are between 0 and 12, their sum can be from 0 to 24.Similarly, b + d can be from 0 to 24.But since a + c = b + d + 12, the minimum value of a + c is 12 (when b + d = 0), and the maximum is 24 (when b + d = 12).So, for each possible value of a + c from 12 to 24, the number of (a, c) pairs is 25 - s, where s = a + c.Similarly, for each s, b + d = s - 12, so the number of (b, d) pairs is (s - 12) + 1 = s - 11.Therefore, the total number of solutions is Σ [s=12 to 24] (25 - s)(s - 11), which is exactly what I did earlier, leading to 455.Alternatively, another substitution: Let me set x = a, y = c, z = b, w = d.Then, the equation is z + w - x - y = -12, which can be rewritten as x + y - z - w = 12.But since x, y, z, w are all non-negative integers, we can think of this as x + y = z + w + 12.But I think the previous approach is more straightforward.Wait, maybe I can model this as a generating function problem.The generating function for a and c is (1 + x + x² + ... + x¹²)², and similarly for b and d, it's (1 + x + x² + ... + x¹²)².But since we have the equation x + y = z + w + 12, the coefficient of x¹² in the product of the generating functions for a + c and b + d would give the number of solutions.Wait, actually, the generating function for a + c is (1 + x + x² + ... + x²⁴), but considering the constraints a, c ≤ 12.Similarly for b + d.But I think this might complicate things more. Maybe it's better to stick with the combinatorial approach.Alternatively, another substitution: Let me set u = a - 12 and v = c - 12. Then, u and v can be from -12 to 0.But I'm not sure if that helps.Wait, going back, I think my initial approach is correct, and the answer is 455.So, I think I'm confident that the number of such polynomials is 455.

💡Alright, so I'm trying to solve this triangle problem. Let me read it again to make sure I understand.In triangle ABC, with angles A ≤ B ≤ C, we're given that the ratio of the sum of the sines of the angles to the sum of the cosines of the angles is equal to the square root of 3. Mathematically, that's:[frac{sin A + sin B + sin C}{cos A + cos B + cos C} = sqrt{3}]And we need to find the value of (sin B + sin 2B).Hmm, okay. So, first, I know that in any triangle, the sum of the angles is π radians (or 180 degrees). So, A + B + C = π. That might come in handy later.Given the ratio, maybe I can express this in terms of some trigonometric identities or perhaps use some substitution to simplify the expression.Let me think about the numerator and the denominator separately.The numerator is (sin A + sin B + sin C), and the denominator is (cos A + cos B + cos C). The ratio is (sqrt{3}).I remember that in triangles, sometimes it's useful to express the sine and cosine of angles in terms of each other or to use sum-to-product identities. Let me see if that can help here.Alternatively, maybe I can consider using the identity for the tangent of an angle since the ratio of sine to cosine is tangent. But here, it's the ratio of sums, not individual terms. So, that might not directly apply.Wait, another thought: maybe I can write this ratio as:[frac{sin A + sin B + sin C}{cos A + cos B + cos C} = sqrt{3}]Which implies:[sin A + sin B + sin C = sqrt{3} (cos A + cos B + cos C)]That's an equation involving sines and cosines of the angles. Maybe I can rearrange this equation to get something more manageable.Let me bring everything to one side:[sin A + sin B + sin C - sqrt{3} cos A - sqrt{3} cos B - sqrt{3} cos C = 0]Hmm, that looks a bit complicated, but perhaps I can factor out the sines and cosines.Wait, another idea: maybe I can use the identity for (sin theta - sqrt{3} cos theta). I recall that this can be written as (2 sin (theta - frac{pi}{3})). Let me verify that.Yes, because:[sin theta - sqrt{3} cos theta = 2 sin left( theta - frac{pi}{3} right)]This is because:[sin theta - sqrt{3} cos theta = 2 left( frac{1}{2} sin theta - frac{sqrt{3}}{2} cos theta right) = 2 sin left( theta - frac{pi}{3} right)]Since (sin (a - b) = sin a cos b - cos a sin b), and (cos frac{pi}{3} = frac{1}{2}), (sin frac{pi}{3} = frac{sqrt{3}}{2}).So, applying this identity to each term in the equation:[(sin A - sqrt{3} cos A) + (sin B - sqrt{3} cos B) + (sin C - sqrt{3} cos C) = 0]Which becomes:[2 sin left( A - frac{pi}{3} right) + 2 sin left( B - frac{pi}{3} right) + 2 sin left( C - frac{pi}{3} right) = 0]Dividing both sides by 2:[sin left( A - frac{pi}{3} right) + sin left( B - frac{pi}{3} right) + sin left( C - frac{pi}{3} right) = 0]Okay, so now we have the sum of these three sine terms equal to zero. Let me denote:[alpha = A - frac{pi}{3}, quad beta = B - frac{pi}{3}, quad gamma = C - frac{pi}{3}]So, the equation becomes:[sin alpha + sin beta + sin gamma = 0]Also, since A + B + C = π, substituting the expressions for α, β, γ:[(A - frac{pi}{3}) + (B - frac{pi}{3}) + (C - frac{pi}{3}) = pi - pi = 0]So, α + β + γ = 0.Therefore, we have:1. (sin alpha + sin beta + sin gamma = 0)2. (alpha + beta + gamma = 0)Hmm, that's interesting. So, the sum of the angles is zero, and the sum of their sines is zero.I wonder if there's a trigonometric identity that relates the sum of sines when the sum of angles is zero.Let me recall that:[sin alpha + sin beta + sin gamma = 4 sin left( frac{alpha + beta}{2} right) sin left( frac{beta + gamma}{2} right) sin left( frac{gamma + alpha}{2} right)]But since α + β + γ = 0, we can write γ = - (α + β). Let me substitute that into the identity.So, γ = - (α + β), so:[sin alpha + sin beta + sin (- (α + β)) = 0]Which simplifies to:[sin alpha + sin beta - sin (α + β) = 0]So, we have:[sin alpha + sin beta = sin (α + β)]Let me write that down:[sin alpha + sin beta = sin (α + β)]Hmm, let's use the sine addition formula on the right side:[sin (α + β) = sin α cos β + cos α sin β]So, substituting back:[sin α + sin β = sin α cos β + cos α sin β]Let me bring all terms to one side:[sin α + sin β - sin α cos β - cos α sin β = 0]Factor terms:[sin α (1 - cos β) + sin β (1 - cos α) = 0]Hmm, that's an interesting equation. Let me see if I can factor this further or find a relationship between α and β.Alternatively, perhaps I can write this as:[sin α (1 - cos β) = - sin β (1 - cos α)]Which implies:[frac{sin α}{1 - cos α} = - frac{sin β}{1 - cos β}]Wait, I recall that (frac{sin θ}{1 - cos θ} = cot frac{θ}{2}). Let me verify that.Yes, because:[frac{sin θ}{1 - cos θ} = frac{2 sin frac{θ}{2} cos frac{θ}{2}}{2 sin^2 frac{θ}{2}} = cot frac{θ}{2}]So, applying this identity:[cot frac{α}{2} = - cot frac{β}{2}]Which implies:[cot frac{α}{2} + cot frac{β}{2} = 0]Hmm, so the sum of these cotangents is zero. That suggests that:[cot frac{α}{2} = - cot frac{β}{2}]Which implies that:[cot frac{α}{2} = cot left( - frac{β}{2} right)]Since cotangent is an odd function, (cot (-x) = - cot x).Therefore, the equation simplifies to:[cot frac{α}{2} = - cot frac{β}{2}]Which suggests that:[frac{α}{2} = - frac{β}{2} + kπ]For some integer k. But since α and β are angles in a triangle adjusted by π/3, they are likely within a range that makes k=0 the only feasible solution.So, simplifying:[frac{α}{2} = - frac{β}{2}]Which implies:[α = - β]So, α = -β.But remember that α + β + γ = 0, so substituting α = -β:[-β + β + γ = 0 implies γ = 0]So, γ = 0.Therefore, going back to the definitions:γ = C - π/3 = 0 ⇒ C = π/3.Wait, but in the problem statement, it's given that A ≤ B ≤ C. So, if C = π/3, then A and B must also be less than or equal to π/3.But in a triangle, the sum of angles is π, so if C = π/3, then A + B = 2π/3.Given that A ≤ B ≤ C = π/3, then A and B must each be less than or equal to π/3, and their sum is 2π/3.This suggests that A = B = π/3 as well, because if A ≤ B ≤ π/3 and A + B = 2π/3, then A must be equal to B, each being π/3.Wait, let me check that.If A ≤ B ≤ C = π/3, and A + B = 2π/3, then since B ≤ π/3, then A = 2π/3 - B ≥ 2π/3 - π/3 = π/3.But since A ≤ B, and A ≥ π/3, and B ≤ π/3, the only possibility is A = B = π/3.Therefore, all angles A, B, C are equal to π/3.So, triangle ABC is equilateral.Wait, but in that case, all angles are π/3, so A = B = C = π/3.Therefore, sin B = sin π/3 = √3/2, and sin 2B = sin(2π/3) = sin(π - π/3) = sin(π/3) = √3/2.Therefore, sin B + sin 2B = √3/2 + √3/2 = √3.So, the value is √3.But wait, let me make sure I didn't make any mistakes in my reasoning.I started by rewriting the given ratio as an equation involving sines and cosines, then used the identity to express each sine minus sqrt(3) cosine as twice the sine of (angle - π/3). That led me to the sum of sines of (A - π/3), (B - π/3), and (C - π/3) equal to zero.Then, by setting α = A - π/3, etc., I found that α + β + γ = 0, and sin α + sin β + sin γ = 0.Through some trigonometric manipulations, I concluded that α = -β, leading to γ = 0, hence C = π/3. Then, considering the ordering of angles, I concluded that A = B = C = π/3, making the triangle equilateral.Therefore, sin B + sin 2B = √3.I think that makes sense. Let me double-check the key steps.1. The ratio given leads to the equation involving sines and cosines.2. Using the identity for sine minus sqrt(3) cosine, I transformed the equation into a sum of sines equal to zero.3. By setting α, β, γ as angle differences from π/3, I found that their sum is zero and their sine sum is zero.4. Through trigonometric identities, I deduced that one of the angles must be zero, leading to C = π/3.5. Considering the angle ordering, all angles must be π/3, making the triangle equilateral.6. Calculated sin B + sin 2B accordingly.Yes, that seems consistent. I don't see any errors in the logic or calculations. So, I'm confident that the answer is √3.**Final Answer**boxed{sqrt{3}}

💡Okay, so I have this problem where I need to find the minimum value of the expression (x + y)/(xy) given that x and y are positive numbers satisfying the equation 4x + y - 1 = 0. Hmm, let me try to figure this out step by step.First, I know that x and y are positive, so both x and y are greater than zero. The equation given is 4x + y = 1. Maybe I can express y in terms of x or vice versa to reduce the number of variables in the expression I need to minimize.Let me solve the equation 4x + y = 1 for y. Subtracting 4x from both sides, I get y = 1 - 4x. Since y has to be positive, 1 - 4x > 0, which means x < 1/4. So x has to be less than 1/4, and since x is positive, x is between 0 and 1/4.Now, I can substitute y = 1 - 4x into the expression (x + y)/(xy). Let's do that:(x + y)/(xy) = (x + (1 - 4x))/(x*(1 - 4x)).Simplify the numerator: x + 1 - 4x = 1 - 3x.So the expression becomes (1 - 3x)/(x*(1 - 4x)).Hmm, that looks a bit complicated. Maybe I can simplify it further or find another approach. Alternatively, I could express x in terms of y from the equation 4x + y = 1. Let's try that.From 4x + y = 1, solving for x gives x = (1 - y)/4. Since x must be positive, (1 - y)/4 > 0, which implies y < 1. So y is between 0 and 1.Substituting x = (1 - y)/4 into the expression (x + y)/(xy):[(1 - y)/4 + y] / [(1 - y)/4 * y].Simplify the numerator: (1 - y)/4 + y = (1 - y + 4y)/4 = (1 + 3y)/4.So the expression becomes [(1 + 3y)/4] / [(1 - y)/4 * y].Simplify the denominator: [(1 - y)/4 * y] = y(1 - y)/4.So now the expression is [(1 + 3y)/4] divided by [y(1 - y)/4], which simplifies to (1 + 3y)/4 * 4/(y(1 - y)).The 4s cancel out, so we have (1 + 3y)/(y(1 - y)).Hmm, that's another expression. Maybe I can work with this. Let me write it as (1 + 3y)/(y - y^2).Alternatively, I can think about using calculus to find the minimum. Since we have a function in terms of one variable, either x or y, we can take the derivative and set it to zero to find critical points.Let me try that. Let's stick with the expression in terms of y: f(y) = (1 + 3y)/(y(1 - y)).First, let's write f(y) as (1 + 3y)/(y - y^2). To find the minimum, I need to find f'(y) and set it equal to zero.Let's compute the derivative f'(y). Using the quotient rule: if f(y) = u/v, then f'(y) = (u'v - uv')/v^2.Here, u = 1 + 3y, so u' = 3.v = y - y^2, so v' = 1 - 2y.So f'(y) = [3*(y - y^2) - (1 + 3y)*(1 - 2y)] / (y - y^2)^2.Let me compute the numerator:First term: 3*(y - y^2) = 3y - 3y^2.Second term: (1 + 3y)*(1 - 2y) = 1*(1 - 2y) + 3y*(1 - 2y) = 1 - 2y + 3y - 6y^2 = 1 + y - 6y^2.So the numerator is (3y - 3y^2) - (1 + y - 6y^2) = 3y - 3y^2 -1 - y + 6y^2.Combine like terms:3y - y = 2y.-3y^2 + 6y^2 = 3y^2.So numerator becomes 2y + 3y^2 -1.Therefore, f'(y) = (3y^2 + 2y -1)/(y - y^2)^2.To find critical points, set numerator equal to zero: 3y^2 + 2y -1 = 0.Solving this quadratic equation: 3y^2 + 2y -1 = 0.Using quadratic formula: y = [-2 ± sqrt(4 + 12)] / 6 = [-2 ± sqrt(16)] /6 = [-2 ±4]/6.So two solutions:y = (-2 + 4)/6 = 2/6 = 1/3.y = (-2 -4)/6 = -6/6 = -1.But y must be positive and less than 1, so y = 1/3 is the critical point.Now, let's check if this is a minimum. We can use the second derivative test or analyze the sign of f'(y) around y = 1/3.Alternatively, since it's the only critical point in the domain (0 < y <1), and the function tends to infinity as y approaches 0 or 1, it's likely that this critical point is a minimum.So, let's compute f(1/3):f(1/3) = (1 + 3*(1/3)) / ( (1/3)*(1 - 1/3) ) = (1 +1) / ( (1/3)*(2/3) ) = 2 / (2/9) = 2 * (9/2) = 9.So, the minimum value is 9.Wait, let me verify this with another method to be sure. Maybe using substitution or AM-GM inequality.Let me try substitution again. From 4x + y =1, y =1 -4x. Then, (x + y)/(xy) = (x +1 -4x)/(x*(1 -4x)) = (1 -3x)/(x -4x^2).Alternatively, maybe express the expression as (x + y)/(xy) = 1/y + 1/x.So, (x + y)/(xy) = 1/x + 1/y.Given that 4x + y =1, so y =1 -4x.So, 1/x + 1/y = 1/x + 1/(1 -4x).Let me denote this as f(x) = 1/x + 1/(1 -4x).Find the minimum of f(x) for 0 < x <1/4.Again, take derivative f'(x) = -1/x^2 + 4/(1 -4x)^2.Set f'(x) =0:-1/x^2 + 4/(1 -4x)^2 =0.So, 4/(1 -4x)^2 =1/x^2.Cross multiplying: 4x^2 = (1 -4x)^2.Take square roots? Or expand.(1 -4x)^2 =1 -8x +16x^2.So, 4x^2 =1 -8x +16x^2.Bring all terms to one side: 4x^2 -16x^2 +8x -1=0.Simplify: -12x^2 +8x -1=0.Multiply both sides by -1: 12x^2 -8x +1=0.Quadratic equation: 12x^2 -8x +1=0.Solutions: x = [8 ± sqrt(64 -48)] /24 = [8 ± sqrt(16)] /24 = [8 ±4]/24.So, x=(8+4)/24=12/24=1/2 or x=(8-4)/24=4/24=1/6.But x must be less than1/4, so x=1/6 is the critical point.Compute f(1/6)=1/(1/6)+1/(1 -4*(1/6))=6 +1/(1 -2/3)=6 +1/(1/3)=6 +3=9.Same result. So the minimum is indeed 9.Alternatively, maybe using AM-GM inequality. Let's see.We have 4x + y =1.We need to minimize (x + y)/(xy) =1/x +1/y.Let me denote a =4x, b=y. Then, a + b =1.We need to express 1/x +1/y in terms of a and b.Since a=4x, x=a/4. Similarly, y=b.So, 1/x +1/y=4/a +1/b.So, we have a + b =1, a>0, b>0.We need to minimize 4/a +1/b.Hmm, maybe use Cauchy-Schwarz or AM-GM.Let me try Cauchy-Schwarz.(4/a +1/b)(a + b) ≥ (2 +1)^2=9.Since a + b=1, so (4/a +1/b)*1 ≥9, which implies 4/a +1/b ≥9.Equality holds when (sqrt(4/a), sqrt(1/b)) is proportional to (sqrt(a), sqrt(b)).Wait, maybe more precise.By Cauchy-Schwarz inequality: (4/a +1/b)(a + b) ≥ (sqrt(4/a *a) + sqrt(1/b *b))^2 = (2 +1)^2=9.Thus, 4/a +1/b ≥9.Equality holds when 4/a^2 =1/b^2, meaning 2/a=1/b, so b=a/2.But a + b=1, so a +a/2=1 => 3a/2=1 => a=2/3, then b=1/3.So, a=2/3, b=1/3.But a=4x=2/3 => x= (2/3)/4=1/6.And b=y=1/3.So, x=1/6, y=1/3, which gives the minimum value of 9.So, all methods lead to the same conclusion.Therefore, the minimum value is 9.**Final Answer**The minimum value is boxed{9}.

💡Alright, so I've got this geometry problem here about a square and a circle. Let me try to visualize it first. There's a square named PQRS with each side measuring 10 feet. A circle is drawn with its center at point M, and this circle passes through two opposite corners of the square, P and R. Additionally, the circle is tangent to one of the sides of the square, specifically side PQ, at a point T. The question is asking for the radius of this circle, and there are multiple-choice options provided.Okay, let's break this down step by step. First, since PQRS is a square, all sides are equal, and all angles are right angles. The square has four vertices: P, Q, R, and S. The circle passes through P and R, which are diagonally opposite corners of the square. That means the circle's center M must lie somewhere along the perpendicular bisector of the diagonal PR. In a square, the diagonals are equal and bisect each other at 90 degrees, so the center of the square, let's call it O, is the intersection point of the diagonals PR and QS.Now, the circle is also tangent to side PQ at point T. Tangency implies that the radius of the circle at the point of tangency is perpendicular to the tangent line. So, the radius MT must be perpendicular to PQ at point T. Since PQ is a horizontal side (assuming the square is oriented with sides parallel to the axes), the radius MT must be a vertical line.Let me try to sketch this mentally. The square PQRS has P at the bottom-left corner, Q at the bottom-right, R at the top-right, and S at the top-left. The diagonal PR goes from P to R, and the center of the square O is the midpoint of PR. The circle passes through P and R, so its center M must lie somewhere along the perpendicular bisector of PR, which is the line QS in the square.But wait, the circle is also tangent to PQ at T. So, the center M must be positioned such that the distance from M to PQ is equal to the radius of the circle. Since MT is perpendicular to PQ, and PQ is horizontal, MT must be vertical. Therefore, the y-coordinate of M must be equal to the radius of the circle.Let me assign coordinates to the square to make this more concrete. Let's place point P at the origin (0,0). Then, since each side is 10 feet, Q would be at (10,0), R at (10,10), and S at (0,10). The center of the square O would then be at (5,5). The diagonal PR goes from (0,0) to (10,10), and its midpoint is indeed (5,5).Now, the circle passes through P(0,0) and R(10,10), and is tangent to PQ at T. Since PQ is the bottom side from (0,0) to (10,0), the point T must lie somewhere on this side. Let's denote T as (t,0), where t is between 0 and 10.The center M of the circle must satisfy two conditions:1. The distance from M to P is equal to the radius.2. The distance from M to R is equal to the radius.3. The distance from M to T is equal to the radius, and MT is perpendicular to PQ.Since MT is perpendicular to PQ, and PQ is horizontal, MT must be vertical. Therefore, the x-coordinate of M must be equal to the x-coordinate of T. So, if T is at (t,0), then M must be at (t, r), where r is the radius of the circle.Now, let's express the distances from M to P and M to R in terms of t and r.The distance from M(t, r) to P(0,0) is:√[(t - 0)^2 + (r - 0)^2] = √(t^2 + r^2)Similarly, the distance from M(t, r) to R(10,10) is:√[(t - 10)^2 + (r - 10)^2]Since both distances are equal to the radius, we can set them equal to each other:√(t^2 + r^2) = √[(t - 10)^2 + (r - 10)^2]Let's square both sides to eliminate the square roots:t^2 + r^2 = (t - 10)^2 + (r - 10)^2Expanding the right side:(t^2 - 20t + 100) + (r^2 - 20r + 100) = t^2 + r^2 - 20t - 20r + 200So, we have:t^2 + r^2 = t^2 + r^2 - 20t - 20r + 200Subtracting t^2 + r^2 from both sides:0 = -20t - 20r + 200Divide both sides by -20:0 = t + r - 10So, t + r = 10Therefore, t = 10 - rNow, we also know that the distance from M(t, r) to T(t, 0) is equal to the radius r. Since MT is vertical, the distance is simply the difference in the y-coordinates:r - 0 = rWhich is consistent, so no new information there.But we also know that the circle passes through P(0,0), so the distance from M(t, r) to P(0,0) is equal to r:√(t^2 + r^2) = rSquaring both sides:t^2 + r^2 = r^2Subtracting r^2 from both sides:t^2 = 0So, t = 0Wait, that can't be right because if t = 0, then T would be at (0,0), which is point P. But the circle is supposed to be tangent to PQ at T, which should be a distinct point from P since it's passing through P. So, this suggests that t cannot be 0.Hmm, maybe I made a mistake in my reasoning. Let's go back.I had the equation t + r = 10 from earlier. And from the distance from M to P, I had √(t^2 + r^2) = r, which led to t^2 + r^2 = r^2, hence t^2 = 0, t = 0. But this contradicts the tangency condition because T would coincide with P.This suggests that my initial assumption about the coordinates might be flawed or that I missed something in the setup.Wait a minute, perhaps I misapplied the distance from M to P. If M is at (t, r), then the distance to P(0,0) is √(t^2 + r^2), which should equal the radius r. So, √(t^2 + r^2) = r implies t^2 + r^2 = r^2, which simplifies to t^2 = 0, so t = 0. But as we saw, this leads to T being at (0,0), which is P, but the circle is supposed to be tangent at T, a different point.This suggests that my coordinate system might not be the best approach here. Maybe I should consider a different coordinate system or approach the problem differently.Alternatively, perhaps I should use the properties of the square and the circle without assigning coordinates immediately.Since the circle passes through P and R, which are diagonally opposite corners of the square, the center M must lie on the perpendicular bisector of PR. In a square, the perpendicular bisector of the diagonal PR is the other diagonal QS. So, M lies somewhere along QS.But the circle is also tangent to side PQ at T. So, the distance from M to PQ must be equal to the radius. Since PQ is a side of the square, and M lies on QS, which is the diagonal from Q(10,0) to S(0,10), we can find the coordinates of M by determining where along QS the distance to PQ equals the radius.Let me try to express this algebraically. Let's denote the coordinates of M as (x, y). Since M lies on QS, which goes from (10,0) to (0,10), the equation of QS is y = -x + 10.The distance from M(x, y) to PQ, which is the line y = 0, is simply the y-coordinate of M, so y = r, where r is the radius.But since M lies on QS, y = -x + 10, so r = -x + 10, which gives x = 10 - r.Now, the distance from M(x, y) to P(0,0) must be equal to r:√(x^2 + y^2) = rSubstituting y = r and x = 10 - r:√((10 - r)^2 + r^2) = rSquaring both sides:(10 - r)^2 + r^2 = r^2Expanding (10 - r)^2:100 - 20r + r^2 + r^2 = r^2Combining like terms:100 - 20r + 2r^2 = r^2Subtracting r^2 from both sides:100 - 20r + r^2 = 0This is a quadratic equation in terms of r:r^2 - 20r + 100 = 0Let's solve for r using the quadratic formula:r = [20 ± √(400 - 400)] / 2r = [20 ± √0] / 2r = 20 / 2r = 10Wait, so r = 10? That would mean the radius is 10 feet, which is the same as the side of the square. But if the radius is 10, then the center M would be at (10 - 10, 10) = (0,10), which is point S. But the circle centered at S(0,10) with radius 10 would pass through P(0,0) and R(10,10), but would it be tangent to PQ at T?Let's check. The distance from M(0,10) to PQ (y=0) is 10, which is the radius, so it would be tangent at T(0,0), which is point P. But the problem states that the circle is tangent to PQ at T, which should be a different point from P. Therefore, r = 10 leads to T coinciding with P, which contradicts the problem statement.This suggests that there might be an error in my approach. Let me reconsider.Perhaps I made a mistake in assuming that the distance from M to PQ is equal to the radius. In reality, since the circle is tangent to PQ at T, the radius at the point of tangency T is perpendicular to PQ. Since PQ is horizontal, the radius MT must be vertical. Therefore, the y-coordinate of M must be equal to the radius, but the x-coordinate of M must be equal to the x-coordinate of T.Wait, that's what I had earlier. So, if T is at (t,0), then M is at (t, r). But earlier, I tried to set the distance from M to P equal to r, which led to t = 0, which is not acceptable.Alternatively, maybe I should consider that the circle passes through P and R, so the center M must be equidistant from P and R, and also at a distance r from T.Let me try to set up the equations again.Let M be at (t, r), since it's vertically above T(t,0).The distance from M to P(0,0) is √(t^2 + r^2) = rSo, √(t^2 + r^2) = rSquaring both sides: t^2 + r^2 = r^2 => t^2 = 0 => t = 0Again, this leads to t = 0, which places T at (0,0), which is P. But the circle is supposed to be tangent at T, a different point.This suggests that my initial assumption that M is at (t, r) might be incorrect. Perhaps M is not directly above T, but rather, the radius MT is perpendicular to PQ, meaning that MT is vertical, so M has the same x-coordinate as T, but the distance from M to T is r, so M is at (t, r).Wait, that's what I had before. So, if M is at (t, r), then the distance from M to P is √(t^2 + r^2) = r, leading to t = 0, which is not acceptable.This seems like a contradiction. Maybe the circle cannot pass through P and R and be tangent to PQ at a point other than P. But the problem states that it is, so there must be a solution.Perhaps I need to consider that the circle passes through P and R, and is tangent to PQ at T, which is not P. Therefore, the center M cannot be at (0,10), but somewhere else.Wait, earlier I considered M lying on QS, the diagonal from Q(10,0) to S(0,10). So, M is on QS, and the distance from M to PQ is r, which is the y-coordinate of M. Also, the distance from M to P is r.But when I tried that, I ended up with r = 10, which led to M being at (0,10), which is S, but that makes T coincide with P.Perhaps I need to consider that the circle passes through P and R, and is tangent to PQ at T, which is not P. Therefore, the center M must be such that it is equidistant from P and R, and the distance from M to PQ is equal to the radius.Let me try to set up the equations again.Let M be at (x, y). Since M is equidistant from P(0,0) and R(10,10), we have:√(x^2 + y^2) = √((x - 10)^2 + (y - 10)^2)Squaring both sides:x^2 + y^2 = (x - 10)^2 + (y - 10)^2Expanding the right side:x^2 - 20x + 100 + y^2 - 20y + 100 = x^2 + y^2 - 20x - 20y + 200Subtracting x^2 + y^2 from both sides:0 = -20x - 20y + 200Dividing by -20:0 = x + y - 10So, x + y = 10This is the equation of the perpendicular bisector of PR, which is QS.Now, the circle is tangent to PQ at T, which is on PQ. Since PQ is the bottom side from (0,0) to (10,0), T must be at (t, 0) for some t between 0 and 10.The radius at the point of tangency T is perpendicular to PQ. Since PQ is horizontal, the radius MT must be vertical. Therefore, the x-coordinate of M must be equal to the x-coordinate of T, so x = t.Also, the distance from M to T is equal to the radius r, which is the vertical distance from M(t, y) to T(t, 0), so r = y.Therefore, y = r, and since x + y = 10, we have x + r = 10, so x = 10 - r.Now, the distance from M(x, y) to P(0,0) is equal to r:√(x^2 + y^2) = rSubstituting x = 10 - r and y = r:√((10 - r)^2 + r^2) = rSquaring both sides:(10 - r)^2 + r^2 = r^2Expanding (10 - r)^2:100 - 20r + r^2 + r^2 = r^2Combining like terms:100 - 20r + 2r^2 = r^2Subtracting r^2 from both sides:100 - 20r + r^2 = 0This is a quadratic equation:r^2 - 20r + 100 = 0Using the quadratic formula:r = [20 ± √(400 - 400)] / 2r = [20 ± 0] / 2r = 10Again, we get r = 10, which implies that M is at (0,10), which is point S, and T is at (0,0), which is P. But the problem states that the circle is tangent to PQ at T, which should be a different point from P. Therefore, this suggests that there is no solution where the circle passes through P and R and is tangent to PQ at a point other than P.But the problem states that such a circle exists, so I must have made a mistake in my reasoning.Wait, perhaps I misapplied the condition that the circle is tangent to PQ at T. If the circle is tangent to PQ at T, then T is the only point of intersection between the circle and PQ. However, the circle already passes through P, which is also on PQ. Therefore, unless T coincides with P, the circle would intersect PQ at two points: P and T. But the problem states that the circle is tangent to PQ at T, implying that T is the only point of intersection. Therefore, T must coincide with P, making the radius 10 feet.But the problem seems to suggest that T is a different point from P, which is why I'm confused.Wait, let me read the problem again carefully: "A circle is drawn with center M such that it passes through vertices P and R and is tangent to side PQ at point T." It doesn't specify that T is different from P, so perhaps T can coincide with P. In that case, the radius would be 10 feet, which is option E.But earlier, when I considered M at (0,10), the circle would pass through P(0,0) and R(10,10), and be tangent to PQ at P(0,0). So, T would be P.However, the problem might be expecting T to be a different point from P, which would mean that my earlier approach is missing something.Alternatively, perhaps the circle is not centered on QS, but somewhere else. Let me consider that.If the circle passes through P and R, its center must lie on the perpendicular bisector of PR, which is QS. So, M must lie on QS. Therefore, M cannot be anywhere else.Given that, and the fact that the circle is tangent to PQ at T, which must be on PQ, and the only way for the circle to be tangent to PQ is if T is P, because any other point would require the circle to intersect PQ at two points: P and T, which contradicts the tangency condition.Therefore, the only possible solution is that T coincides with P, making the radius 10 feet.But the problem seems to imply that T is a different point, so perhaps I'm misunderstanding the problem.Wait, maybe the circle is tangent to PQ at T, which is not P, but since the circle already passes through P, which is on PQ, the only way for the circle to be tangent to PQ is if T is P. Otherwise, it would intersect PQ at two points: P and T, which would not be a tangency.Therefore, the only possible solution is that T is P, and the radius is 10 feet, which is option E.However, the answer choices include 5√6, which is approximately 12.247, which is larger than 10, which seems impossible because the diagonal of the square is 10√2 ≈ 14.142, so a radius of 5√6 ≈ 12.247 would extend beyond the square.Wait, but if the center M is outside the square, perhaps the radius can be larger than 10. Let me consider that.If M is outside the square, then the circle can have a larger radius. Let me try to visualize this.If M is above the square, then the circle can pass through P and R, and be tangent to PQ at some point T between P and Q.Let me try to set up the equations again, considering that M might be outside the square.Let M be at (t, r), with t between 0 and 10, and r > 10, since it's above the square.The distance from M(t, r) to P(0,0) is √(t^2 + r^2) = radius.The distance from M(t, r) to R(10,10) is √((t - 10)^2 + (r - 10)^2) = radius.Since both distances are equal to the radius, we have:√(t^2 + r^2) = √((t - 10)^2 + (r - 10)^2)Squaring both sides:t^2 + r^2 = (t - 10)^2 + (r - 10)^2Expanding:t^2 + r^2 = t^2 - 20t + 100 + r^2 - 20r + 100Simplifying:0 = -20t - 20r + 20020t + 20r = 200t + r = 10So, t = 10 - rNow, the distance from M(t, r) to T(t, 0) is r, since MT is vertical.But T is on PQ, which is from (0,0) to (10,0). So, t must be between 0 and 10.Given that t = 10 - r, and t must be between 0 and 10, then r must be between 0 and 10. But earlier, I considered r > 10, which would make t negative, which is not possible since t must be between 0 and 10.Therefore, M must be inside the square, with r ≤ 10.But earlier, when I tried r = 10, M was at (0,10), which is S, and T was at (0,0), which is P.So, perhaps the only solution is r = 10, with T coinciding with P.But the problem states that the circle is tangent to PQ at T, which could be P, so maybe that's acceptable.However, the answer choices include 5√6, which is approximately 12.247, which is larger than 10, so perhaps I need to reconsider.Wait, maybe I made a mistake in assuming that M lies on QS. Let me think again.The perpendicular bisector of PR is QS, so M must lie on QS. Therefore, M is on QS, and the distance from M to PQ is r, which is the y-coordinate of M.But if M is on QS, which goes from (10,0) to (0,10), then M can be anywhere along that line, inside or outside the square.Wait, if M is outside the square, say above it, then t = 10 - r would be negative, which would place T at (t,0) with t negative, which is outside the square. But T must be on PQ, which is from (0,0) to (10,0). Therefore, t must be between 0 and 10, so r must be between 0 and 10.Therefore, M must be inside the square, with r ≤ 10.But then, as before, the only solution is r = 10, with M at (0,10), which is S, and T at (0,0), which is P.Therefore, the radius must be 10 feet, which is option E.But the problem seems to suggest that T is a different point from P, which is why I'm confused.Wait, perhaps the circle is not centered on QS, but somewhere else. But that contradicts the fact that M must lie on the perpendicular bisector of PR, which is QS.Alternatively, perhaps the circle is not passing through P and R, but through P and another point. Wait, no, the problem says it passes through P and R.Wait, maybe I'm misinterpreting the problem. Let me read it again: "A circle is drawn with center M such that it passes through vertices P and R and is tangent to side PQ at point T."So, it passes through P and R, and is tangent to PQ at T. It doesn't say that T is different from P, so T could be P.Therefore, the radius is 10 feet, which is option E.But earlier, I thought that the answer might be 5√6, which is approximately 12.247, but that would require M to be outside the square, which is not possible because t would be negative, placing T outside PQ.Therefore, the correct answer must be 10 feet, option E.However, I'm still unsure because the problem might be expecting T to be a different point from P, but mathematically, it seems that the only solution is T coinciding with P.Alternatively, perhaps I made a mistake in my calculations. Let me try to solve the quadratic equation again.From earlier, we had:r^2 - 20r + 100 = 0Using the quadratic formula:r = [20 ± √(400 - 400)] / 2r = [20 ± 0] / 2r = 10So, the only solution is r = 10.Therefore, the radius must be 10 feet, which is option E.I think I've exhausted all possibilities, and the only solution that satisfies all conditions is r = 10 feet.

💡To determine the sum of the digits of the square of (222222222), we can observe a pattern in the squares of numbers consisting of repeated digits of 2.1. **Identify the Pattern**: - (2^2 = 4) - (22^2 = 484) - (222^2 = 49284) - (2222^2 = 4944284) From these examples, we notice that the digits of the squares of numbers with repeated digits of 2 follow a specific pattern.2. **Generalize the Pattern**: Let (Y = 222cdots222) (with (n) digits of 2). The square of (Y) results in a number whose digits increase to a certain point and then decrease. This pattern is consistent for numbers with repeated digits of 2.3. **Sum of the Digits**: The sum of the digits of (Y^2) can be approximated by considering the pattern observed. For a number with (n) digits of 2, the sum of the digits of its square is given by: [ 2 times (1 + 2 + 3 + cdots + n + (n-1) + (n-2) + cdots + 1) ]4. **Use the Sum Formula**: The sum of the first (n) natural numbers is: [ frac{n(n+1)}{2} ] Therefore, the total sum of the digits is: [ 2 times left( frac{n(n+1)}{2} + frac{(n-1)n}{2} right) = n^2 times 2 ]5. **Calculate the Total Sum**: For (n = 9) (since there are nine 2s in (222222222)): [ 2 times (9^2) = 2 times 81 = 162 ]6. **Conclusion**: The sum of the digits of the square of (222222222) is (162).The final answer is (boxed{D}).

💡Alright, so I have this problem here: Compute (3 cos 30° + 3i sin 30°)^4 and enter the answer in rectangular form. Hmm, okay. I remember that when dealing with complex numbers in trigonometric form, there's something called De Moivre's Theorem that can help raise them to powers. Let me try to recall how that works.First, the complex number given is 3 cos 30° + 3i sin 30°. I notice that both the cosine and sine terms have the same coefficient, which is 3. That makes me think that this is a complex number in polar form, right? So, in general, a complex number can be written as r(cos θ + i sin θ), where r is the modulus and θ is the argument. In this case, r is 3 and θ is 30°. So, this is already in polar form, which is good because De Moivre's Theorem applies directly to this form.De Moivre's Theorem states that for any complex number in polar form, (r(cos θ + i sin θ))^n = r^n (cos nθ + i sin nθ). So, if I apply this theorem, I can raise the modulus r to the power of n and multiply the angle θ by n. That should make the computation easier than expanding the binomial directly, which would be quite tedious for the fourth power.So, let's apply De Moivre's Theorem here. The modulus r is 3, and we're raising it to the fourth power. So, 3^4 is... let me calculate that. 3 squared is 9, and 9 squared is 81. So, 3^4 is 81. Got that part.Next, the angle θ is 30°, and we need to multiply that by 4 because we're raising it to the fourth power. So, 30° times 4 is 120°. So, the angle becomes 120°. Therefore, according to De Moivre's Theorem, (3 cos 30° + 3i sin 30°)^4 should be equal to 81 (cos 120° + i sin 120°).Now, I need to convert this back into rectangular form, which is a + bi. To do that, I need to compute the cosine and sine of 120° and then multiply them by 81.Let me recall the values of cosine and sine for 120°. I know that 120° is in the second quadrant, where cosine is negative and sine is positive. Also, 120° is equal to 180° - 60°, so it's related to the 60° reference angle. I remember that cos 60° is 0.5 and sin 60° is (√3)/2. Therefore, cos 120° should be -0.5, and sin 120° should be (√3)/2.So, substituting these values back into the expression, we get:81 (cos 120° + i sin 120°) = 81 (-0.5 + i (√3)/2)Now, let's compute this multiplication. First, multiply 81 by -0.5. 81 times 0.5 is 40.5, so 81 times -0.5 is -40.5. Next, multiply 81 by (√3)/2. 81 divided by 2 is 40.5, so 40.5 times √3 is 40.5√3. Therefore, the expression becomes:-40.5 + i 40.5√3So, in rectangular form, the complex number is -40.5 + 40.5√3 i.Wait, let me double-check my calculations to make sure I didn't make any mistakes. Starting with the modulus: 3^4 is indeed 81. The angle: 30° times 4 is 120°, correct. Cosine of 120° is -0.5, sine is √3/2, that's right. Multiplying 81 by -0.5 gives -40.5, and 81 times √3/2 is 40.5√3. So, putting it all together, the rectangular form is -40.5 + 40.5√3 i.I think that's correct. But just to be thorough, maybe I should verify using another method, like converting the original complex number to rectangular form first and then raising it to the fourth power.Let's see. The original complex number is 3 cos 30° + 3i sin 30°. Let's compute cos 30° and sin 30°. Cos 30° is √3/2, which is approximately 0.8660, and sin 30° is 0.5. So, substituting these values, the complex number becomes:3*(√3/2) + 3i*(0.5) = (3√3)/2 + (3/2)iCalculating the numerical values: (3√3)/2 is approximately (3*1.732)/2 ≈ 5.196/2 ≈ 2.598, and (3/2) is 1.5. So, the complex number is approximately 2.598 + 1.5i.Now, let's raise this to the fourth power. That might be a bit tedious, but let's try it step by step.First, compute (2.598 + 1.5i)^2.Using the formula (a + b)^2 = a^2 + 2ab + b^2, where a = 2.598 and b = 1.5i.So, (2.598)^2 + 2*(2.598)*(1.5i) + (1.5i)^2.Calculating each term:(2.598)^2 ≈ 7.02*(2.598)*(1.5i) ≈ 2*2.598*1.5i ≈ 7.794i(1.5i)^2 = (1.5)^2 * i^2 = 2.25*(-1) = -2.25Adding these together: 7.0 + 7.794i - 2.25 = (7.0 - 2.25) + 7.794i = 4.75 + 7.794iSo, (2.598 + 1.5i)^2 ≈ 4.75 + 7.794iNow, we need to square this result to get the fourth power.So, (4.75 + 7.794i)^2.Again, using the formula (a + b)^2 = a^2 + 2ab + b^2, where a = 4.75 and b = 7.794i.Calculating each term:(4.75)^2 = 22.56252*(4.75)*(7.794i) ≈ 2*4.75*7.794i ≈ 73.743i(7.794i)^2 = (7.794)^2 * i^2 ≈ 60.75*(-1) ≈ -60.75Adding these together: 22.5625 + 73.743i - 60.75 ≈ (22.5625 - 60.75) + 73.743i ≈ -38.1875 + 73.743iHmm, so according to this method, the result is approximately -38.1875 + 73.743i.Wait a minute, that's different from the previous result of -40.5 + 40.5√3 i. Let me compute 40.5√3 to see what that is approximately.40.5 * 1.732 ≈ 40.5 * 1.732 ≈ 69.966So, the previous result was approximately -40.5 + 69.966i, which is close to -40.5 + 70i, while the other method gave me approximately -38.1875 + 73.743i.These are not the same. There must be a mistake somewhere. Let me check my calculations again.Starting with the first method: using De Moivre's Theorem, I got -40.5 + 40.5√3 i, which is approximately -40.5 + 70i.In the second method, converting to rectangular form first and then squaring twice, I got approximately -38.1875 + 73.743i.These discrepancies suggest that I made an error in one of the methods. Let me check the second method more carefully.First, converting 3 cos 30° + 3i sin 30° to rectangular form:cos 30° = √3/2 ≈ 0.8660sin 30° = 0.5So, 3 cos 30° ≈ 3 * 0.8660 ≈ 2.5983 sin 30° ≈ 3 * 0.5 = 1.5So, the rectangular form is approximately 2.598 + 1.5i, which is correct.Now, squaring this: (2.598 + 1.5i)^2Let me compute this more accurately.First, (2.598)^2: 2.598 * 2.598Let me compute 2.598 * 2.598:2 * 2.598 = 5.1960.598 * 2.598 ≈ 1.554So, total is approximately 5.196 + 1.554 ≈ 6.75Wait, that's different from my previous calculation where I thought it was 7.0. Hmm, maybe I was approximating too much.Similarly, 2*(2.598)*(1.5i) = 2*2.598*1.5i = 7.794i, which seems correct.(1.5i)^2 = -2.25, correct.So, adding them together: 6.75 + 7.794i - 2.25 = 4.5 + 7.794iWait, so previously I thought (2.598)^2 was 7.0, but actually it's approximately 6.75. So, that changes the result.So, the square is approximately 4.5 + 7.794i.Now, squaring this result: (4.5 + 7.794i)^2Compute each term:(4.5)^2 = 20.252*(4.5)*(7.794i) = 2*4.5*7.794i ≈ 69.546i(7.794i)^2 = (7.794)^2 * i^2 ≈ 60.75*(-1) ≈ -60.75Adding these together: 20.25 + 69.546i - 60.75 ≈ (20.25 - 60.75) + 69.546i ≈ -40.5 + 69.546iAh, now this is closer to the result from the first method. So, approximately -40.5 + 69.546i, which is very close to -40.5 + 40.5√3 i, since 40.5√3 ≈ 69.966.The slight difference is due to rounding errors in the approximate calculations. So, both methods agree when done accurately.Therefore, the correct rectangular form is -40.5 + 40.5√3 i.I think I made a mistake in my initial squaring step when I approximated (2.598)^2 as 7.0 instead of the more accurate 6.75, which led to a discrepancy. After correcting that, both methods give consistent results.So, to summarize, using De Moivre's Theorem is more straightforward and less error-prone, especially for higher powers, as it avoids the need for multiple squaring steps which can introduce rounding errors or calculation mistakes.Another way to think about it is that the original complex number is 3 (cos 30° + i sin 30°), which is 3 cis 30°, where "cis" is shorthand for cos + i sin. Raising this to the fourth power gives 3^4 cis (4*30°) = 81 cis 120°, which is 81 (cos 120° + i sin 120°). As we calculated earlier, cos 120° is -0.5 and sin 120° is √3/2, so multiplying these by 81 gives the rectangular form.I think I've covered all the steps and checked my work to ensure accuracy. It's always good to verify using different methods to catch any potential errors. In this case, both methods led to the same conclusion once I corrected my initial approximation mistake.So, the final answer in rectangular form is -40.5 + 40.5√3 i.