🧠:<think>Alright, so I've got this problem about Carl, the classical composer, who's trying to model the complexity of music using some quadratic functions. There are two sub-problems here, and I need to tackle them step by step. Let me start with the first one.Sub-problem 1:Carl defines the complexity ( C(x, y) ) as a quadratic function:[ C(x, y) = ax^2 + bxy + cy^2 + dx + ey + f ]He gives that for his classical composition, ( x = 8 ) and ( y = 5 ), the complexity is 150. So plugging those values in:[ 150 = a(8)^2 + b(8)(5) + c(5)^2 + d(8) + e(5) + f ][ 150 = 64a + 40b + 25c + 8d + 5e + f ]He also mentions that the most complex film score will have a complexity of at most 70% of his composition's complexity. So 70% of 150 is 105. Therefore, for a film score with ( x = 6 ) and ( y = 4 ), the complexity should be ≤ 105.So plugging ( x = 6 ) and ( y = 4 ) into the complexity function:[ C(6, 4) = a(6)^2 + b(6)(4) + c(4)^2 + d(6) + e(4) + f ][ C(6, 4) = 36a + 24b + 16c + 6d + 4e + f ]And this should be ≤ 105.Additionally, Carl gives some constraints on the constants: ( a = c ) and ( b = 2a ). So let's substitute these into our equations.First, let's rewrite the complexity function with these constraints:Since ( a = c ) and ( b = 2a ), the function becomes:[ C(x, y) = ax^2 + 2a xy + a y^2 + dx + ey + f ][ C(x, y) = a(x^2 + 2xy + y^2) + dx + ey + f ]Notice that ( x^2 + 2xy + y^2 = (x + y)^2 ), so:[ C(x, y) = a(x + y)^2 + dx + ey + f ]That's a nice simplification. Now, let's plug in the values for Carl's composition:For ( x = 8 ), ( y = 5 ):[ 150 = a(8 + 5)^2 + d(8) + e(5) + f ][ 150 = a(13)^2 + 8d + 5e + f ][ 150 = 169a + 8d + 5e + f ] --- Equation (1)For the film score, ( x = 6 ), ( y = 4 ):[ C(6, 4) = a(6 + 4)^2 + d(6) + e(4) + f ][ C(6, 4) = a(10)^2 + 6d + 4e + f ][ C(6, 4) = 100a + 6d + 4e + f ]And this must be ≤ 105. So:[ 100a + 6d + 4e + f ≤ 105 ] --- Inequality (2)Now, we have Equation (1):169a + 8d + 5e + f = 150And Inequality (2):100a + 6d + 4e + f ≤ 105We need to find the maximum possible values of a, b, c, d, e, f. Since b and c are dependent on a, and a is a variable here, we can focus on finding a, d, e, f.Let me subtract Inequality (2) from Equation (1) to see what we get.Equation (1) - Inequality (2):(169a - 100a) + (8d - 6d) + (5e - 4e) + (f - f) ≥ 150 - 105Wait, actually, since Inequality (2) is ≤ 105, subtracting it from Equation (1) would give:169a + 8d + 5e + f - (100a + 6d + 4e + f) ≥ 150 - 105Simplify:69a + 2d + e ≥ 45So,69a + 2d + e ≥ 45 --- Inequality (3)Our goal is to maximize a, d, e, f. But we have multiple variables here. Let me think about how to approach this.Since we're trying to maximize the constants, perhaps we can express d and e in terms of a, and then find the maximum a such that Inequality (3) is satisfied, while also ensuring that Inequality (2) holds.Alternatively, since we have two equations (Equation 1 and Inequality 2), and three variables (a, d, e, f), but f can be expressed in terms of a, d, e from Equation (1):From Equation (1):f = 150 - 169a - 8d - 5ePlugging this into Inequality (2):100a + 6d + 4e + (150 - 169a - 8d - 5e) ≤ 105Simplify:100a + 6d + 4e + 150 - 169a - 8d - 5e ≤ 105Combine like terms:(100a - 169a) + (6d - 8d) + (4e - 5e) + 150 ≤ 105-69a - 2d - e + 150 ≤ 105Bring constants to the right:-69a - 2d - e ≤ -45Multiply both sides by -1 (which reverses the inequality):69a + 2d + e ≥ 45Which is the same as Inequality (3). So, that didn't give us new information.So, we have:69a + 2d + e ≥ 45And from Equation (1):f = 150 - 169a - 8d - 5eWe need to find the maximum possible a, d, e, f such that 69a + 2d + e ≥ 45 and f is as large as possible? Wait, but we need to maximize all constants. Hmm, this is a bit tricky.Wait, actually, the problem says "solve for the maximum possible values of a, b, c, d, e, and f" with the given constraints. So, perhaps we need to maximize each constant individually, but they are interdependent.Wait, maybe another approach: Since we need to maximize a, d, e, f, perhaps we can set 69a + 2d + e = 45, because if we set it equal, that would give us the minimal possible left-hand side, allowing a, d, e to be as large as possible. Wait, but 69a + 2d + e is greater than or equal to 45, so to maximize a, d, e, we might need to set 69a + 2d + e as small as possible, which is 45.Wait, that might be the case. Let me think.If we set 69a + 2d + e = 45, then we can solve for one variable in terms of others.But we also have f expressed in terms of a, d, e.So, let me try to express e from Inequality (3):e = 45 - 69a - 2dBut since e is a constant, it must be a real number. So, 45 - 69a - 2d must be ≥ something? Wait, no, e can be any real number, positive or negative. But in the context of complexity, maybe the constants should be positive? Hmm, the problem doesn't specify, so perhaps they can be any real numbers.But in order to maximize a, d, e, f, we might need to set e as large as possible, but e is dependent on a and d.Wait, perhaps another approach: Let's consider that we have two equations:1. 169a + 8d + 5e + f = 1502. 100a + 6d + 4e + f ≤ 105We can subtract the second inequality from the first equation:(169a - 100a) + (8d - 6d) + (5e - 4e) + (f - f) ≥ 150 - 105Which simplifies to:69a + 2d + e ≥ 45So, 69a + 2d + e must be at least 45.To maximize a, d, e, f, perhaps we can set 69a + 2d + e = 45, and then express e in terms of a and d: e = 45 - 69a - 2dThen plug this into Equation (1):169a + 8d + 5*(45 - 69a - 2d) + f = 150Simplify:169a + 8d + 225 - 345a - 10d + f = 150Combine like terms:(169a - 345a) + (8d - 10d) + 225 + f = 150-176a - 2d + 225 + f = 150Then,-176a - 2d + f = -75So,f = 176a + 2d - 75Now, we have e = 45 - 69a - 2d and f = 176a + 2d - 75Our goal is to maximize a, d, e, f.But how? We have two expressions for e and f in terms of a and d. To maximize a, d, e, f, perhaps we can set e and f as large as possible, but they are dependent on a and d.Wait, but e = 45 - 69a - 2d. To maximize e, we need to minimize 69a + 2d. But since 69a + 2d = 45 - e, and we want e to be as large as possible, that would require 69a + 2d to be as small as possible. But 69a + 2d must be ≥ 45 - e, but e can be any value. Hmm, this is getting a bit convoluted.Alternatively, perhaps we can treat a and d as variables and express e and f in terms of them, then find the maximum values.But without additional constraints, it's difficult to find unique maximum values. Maybe the problem expects us to find the maximum possible a, given the constraints, and then express d, e, f in terms of a.Wait, let's think about this differently. Since we need to maximize a, d, e, f, perhaps we can set d and e to be as large as possible, but they are constrained by the equations.Alternatively, perhaps we can set d and e to zero to maximize a and f.Wait, if we set d = 0 and e = 0, then from Inequality (3):69a ≥ 45 => a ≥ 45/69 ≈ 0.652But from Equation (1):169a + f = 150 => f = 150 - 169aIf a is as large as possible, f would be as small as possible, but we want to maximize f as well. So this approach might not work.Alternatively, perhaps we can set d and e to be as large as possible to allow a to be as large as possible.Wait, I'm getting stuck here. Maybe another approach: Let's assume that the maximum occurs when the inequality becomes an equality, i.e., 100a + 6d + 4e + f = 105.So, we have:169a + 8d + 5e + f = 150 --- Equation (1)100a + 6d + 4e + f = 105 --- Equation (2)Now, subtract Equation (2) from Equation (1):69a + 2d + e = 45 --- Equation (3)So now, we have two equations:Equation (1): 169a + 8d + 5e + f = 150Equation (2): 100a + 6d + 4e + f = 105Equation (3): 69a + 2d + e = 45Now, we can solve this system of equations.From Equation (3): e = 45 - 69a - 2dPlug e into Equation (1):169a + 8d + 5*(45 - 69a - 2d) + f = 150Simplify:169a + 8d + 225 - 345a - 10d + f = 150Combine like terms:(169a - 345a) + (8d - 10d) + 225 + f = 150-176a - 2d + 225 + f = 150So,-176a - 2d + f = -75Thus,f = 176a + 2d - 75Now, plug e and f into Equation (2):100a + 6d + 4*(45 - 69a - 2d) + (176a + 2d - 75) = 105Simplify:100a + 6d + 180 - 276a - 8d + 176a + 2d - 75 = 105Combine like terms:(100a - 276a + 176a) + (6d - 8d + 2d) + (180 - 75) = 105(0a) + (0d) + 105 = 105So, 105 = 105, which is always true. This means that our system is consistent, and we have infinitely many solutions depending on a and d.So, to find the maximum possible values of a, d, e, f, we need to express them in terms of a and d, but without additional constraints, we can't find unique maximums. However, perhaps the problem expects us to express the constants in terms of a, given that a can be any value as long as the other constraints are satisfied.Wait, but the problem says "solve for the maximum possible values of a, b, c, d, e, and f". So, maybe we need to find the maximum a such that all other constants are also maximized. But how?Alternatively, perhaps we can set d and e to be as large as possible, but they are dependent on a. Wait, but without constraints on d and e, they can be any values, so a could be as large as possible, but then f would have to compensate.Wait, perhaps another approach: Let's express everything in terms of a.From Equation (3): e = 45 - 69a - 2dFrom Equation (1): f = 150 - 169a - 8d - 5eSubstitute e into f:f = 150 - 169a - 8d - 5*(45 - 69a - 2d)= 150 - 169a - 8d - 225 + 345a + 10d= (150 - 225) + (-169a + 345a) + (-8d + 10d)= -75 + 176a + 2dSo, f = 176a + 2d - 75Now, since we need to maximize a, d, e, f, perhaps we can set d to be as large as possible. But without constraints on d, it can be any value, which would make a and f also dependent on d.Wait, but if we set d to be as large as possible, then a and f would also increase. However, e = 45 - 69a - 2d. If d increases, e decreases. So, to maximize e, we need to minimize d.Wait, this is getting complicated. Maybe the problem expects us to express the constants in terms of a, with a being a parameter, but I'm not sure.Alternatively, perhaps the maximum occurs when the film score's complexity is exactly 105, which would make the inequality an equality. So, we can solve the system as if 100a + 6d + 4e + f = 105.Which is what we did earlier, leading to the conclusion that the system is consistent with infinitely many solutions.But since we need to find the maximum possible values, perhaps we can set d and e to be as large as possible, but that would require a to be as large as possible as well, but e is dependent on a and d.Wait, maybe we can set d = 0 to see what happens.If d = 0, then from Equation (3):69a + e = 45 => e = 45 - 69aFrom Equation (1):169a + 5e + f = 150Substitute e:169a + 5*(45 - 69a) + f = 150169a + 225 - 345a + f = 150-176a + f = -75f = 176a - 75Now, from Equation (2):100a + 4e + f = 105Substitute e and f:100a + 4*(45 - 69a) + (176a - 75) = 105100a + 180 - 276a + 176a - 75 = 105(100a - 276a + 176a) + (180 - 75) = 1050a + 105 = 105Which is true, so d = 0 is acceptable.Now, to maximize a, d, e, f, with d = 0:e = 45 - 69af = 176a - 75We need to ensure that e is as large as possible. Since e = 45 - 69a, to maximize e, we need to minimize a. But a is a coefficient in the complexity function, and if a is too small, f might become negative, but the problem doesn't specify constraints on the constants being positive.Wait, but if we set a to be as large as possible, e would become negative, but f would increase.Alternatively, perhaps the problem expects a, b, c, d, e, f to be positive. If that's the case, then we can set constraints:e = 45 - 69a ≥ 0 => 45 ≥ 69a => a ≤ 45/69 ≈ 0.652Similarly, f = 176a - 75 ≥ 0 => 176a ≥ 75 => a ≥ 75/176 ≈ 0.426So, a must be between approximately 0.426 and 0.652.To maximize a, set a = 45/69 = 15/23 ≈ 0.652Then,e = 45 - 69*(15/23) = 45 - (69*15)/23 = 45 - (3*15) = 45 - 45 = 0f = 176*(15/23) - 75 = (2640/23) - 75 ≈ 114.78 - 75 = 39.78But since we're dealing with exact fractions, let's compute:176*(15/23) = (176/23)*15 = (8*22/23)*15 = (8*15)*(22/23) = 120*(22/23) = 2640/23So, f = 2640/23 - 75 = 2640/23 - 1725/23 = 915/23 ≈ 39.78So, with a = 15/23, d = 0, e = 0, f = 915/23But wait, e = 0 here. Is that acceptable? The problem doesn't specify that e has to be positive, just that it's a constant. So, perhaps this is acceptable.But let's check if this satisfies all conditions.From Equation (1):169a + 8d + 5e + f = 150Plug in a = 15/23, d = 0, e = 0, f = 915/23:169*(15/23) + 0 + 0 + 915/23 = (2535/23) + (915/23) = (2535 + 915)/23 = 3450/23 ≈ 150Which is correct.From Equation (2):100a + 6d + 4e + f = 105100*(15/23) + 0 + 0 + 915/23 = (1500/23) + (915/23) = 2415/23 = 105Which is correct.So, this solution satisfies both equations.But is this the maximum possible a? Yes, because if we increase a beyond 15/23, e becomes negative, which might not be desired if we assume constants should be non-negative. But the problem doesn't specify, so technically, a could be larger, making e negative, but f would increase.However, if we consider that constants can be negative, then a can be as large as possible, but then e would be negative, and f would be larger.But since the problem asks for the maximum possible values, perhaps we can let a be as large as possible, making e as negative as possible, but f as large as possible.But without constraints, a can be any real number, so technically, a can approach infinity, making e approach negative infinity, and f approach positive infinity. But that doesn't make sense in the context of the problem, as complexity can't be negative or infinitely large.Therefore, perhaps the problem expects us to consider positive constants. So, with a ≤ 15/23, e ≥ 0, and f ≥ 0.So, the maximum a is 15/23, with d = 0, e = 0, f = 915/23.But let's check if d can be positive to allow a to be larger.Wait, if we set d > 0, then from Equation (3):69a + 2d + e = 45If d increases, then 69a + e decreases, allowing a to be larger if e is allowed to be negative.But if we allow e to be negative, then a can be larger than 15/23.But again, without constraints on e being positive, a can be increased indefinitely by increasing d and allowing e to be negative.But in the context of the problem, perhaps e should be positive, as it's a coefficient in the complexity function, which might represent a positive contribution. So, assuming e ≥ 0, then a ≤ 15/23.Therefore, the maximum a is 15/23, with d = 0, e = 0, f = 915/23.But let's check if d can be positive while keeping e positive.Suppose we set d = k, then e = 45 - 69a - 2kTo keep e ≥ 0:45 - 69a - 2k ≥ 0 => 69a + 2k ≤ 45We also have f = 176a + 2k - 75To keep f ≥ 0:176a + 2k ≥ 75So, we have:69a + 2k ≤ 45 --- (A)176a + 2k ≥ 75 --- (B)Subtract (A) from (B):(176a + 2k) - (69a + 2k) ≥ 75 - 45107a ≥ 30a ≥ 30/107 ≈ 0.280So, a must be at least 30/107.But we also have from (A):69a ≤ 45 - 2kBut since k can be any positive number, as long as 69a + 2k ≤ 45.But to maximize a, we can set k as small as possible, which is k = 0, leading us back to a = 15/23 ≈ 0.652.Therefore, the maximum a is 15/23, with d = 0, e = 0, f = 915/23.So, summarizing:a = 15/23b = 2a = 30/23c = a = 15/23d = 0e = 0f = 915/23But let me double-check these values.Plugging into Equation (1):169*(15/23) + 8*0 + 5*0 + 915/23 = (2535/23) + (915/23) = 3450/23 = 150, which is correct.Plugging into Equation (2):100*(15/23) + 6*0 + 4*0 + 915/23 = (1500/23) + (915/23) = 2415/23 = 105, which is correct.So, this solution works.Sub-problem 2:Now, Carl's assistant suggests adding a third dimension, emotional depth ( z ), so the complexity function becomes:[ C(x, y, z) = ax^2 + bxy + cy^2 + dz^2 + exz + fyz + gx + hy + iz + j ]Given that Carl's classical composition has ( x = 8 ), ( y = 5 ), ( z = 7 ), and complexity 250. So:[ 250 = a(8)^2 + b(8)(5) + c(5)^2 + d(7)^2 + e(8)(7) + f(5)(7) + g(8) + h(5) + i(7) + j ]Simplify:[ 250 = 64a + 40b + 25c + 49d + 56e + 35f + 8g + 5h + 7i + j ]We are told that ( h = i = j = 0 ), so the equation simplifies to:[ 250 = 64a + 40b + 25c + 49d + 56e + 35f + 8g ]Additionally, we need to maintain the conditions from sub-problem 1 for ( a, b, c ). From sub-problem 1, we have:a = 15/23, b = 30/23, c = 15/23, d = 0, e = 0, f = 915/23Wait, but in sub-problem 1, d, e, f were specific constants. However, in sub-problem 2, the function has new constants d, e, f, g, h, i, j. But the problem says "maintaining the conditions from sub-problem 1 for a, b, c". So, a, b, c remain the same as in sub-problem 1, but d, e, f, g, h, i, j are new constants to be determined.Wait, let me read the problem again:"Carl's assistant suggests that incorporating a third dimension, the emotional depth ( z ), could offer a more comprehensive view of composition complexity. The new complexity function becomes:[ C(x, y, z) = ax^2 + bxy + cy^2 + dz^2 + exz + fyz + gx + hy + iz + j ]If Carl's classical composition with ( x = 8 ), ( y = 5 ), and ( z = 7 ) results in a complexity of 250, determine the new constants ( d, e, f, ) and ( g ) given that ( h = i = j = 0 ), while maintaining the conditions from sub-problem 1 for ( a, b, c )."So, the new constants are d, e, f, g, h, i, j, but h = i = j = 0, so we only need to find d, e, f, g.From sub-problem 1, we have a = 15/23, b = 30/23, c = 15/23.So, plugging these into the equation for the classical composition:250 = 64*(15/23) + 40*(30/23) + 25*(15/23) + 49d + 56e + 35f + 8gLet's compute each term:64*(15/23) = 960/23 ≈ 41.73940*(30/23) = 1200/23 ≈ 52.17425*(15/23) = 375/23 ≈ 16.296So, sum of these terms:960/23 + 1200/23 + 375/23 = (960 + 1200 + 375)/23 = 2535/23 ≈ 110.217So, the equation becomes:250 = 2535/23 + 49d + 56e + 35f + 8gConvert 250 to 23 denominator:250 = 250*23/23 = 5750/23So,5750/23 = 2535/23 + 49d + 56e + 35f + 8gSubtract 2535/23:(5750 - 2535)/23 = 49d + 56e + 35f + 8g3215/23 = 49d + 56e + 35f + 8gSimplify 3215/23:3215 ÷ 23 = 139.7826... Wait, let me compute 23*139 = 3197, 23*140 = 3220, so 3215 is 3220 - 5 = 23*140 - 5, so 3215/23 = 140 - 5/23 ≈ 139.7826But perhaps it's better to keep it as 3215/23.So,49d + 56e + 35f + 8g = 3215/23Now, we need to find d, e, f, g such that this equation holds. However, we have only one equation with four variables, so we need more information or constraints.Wait, the problem says "determine the new constants d, e, f, and g". It doesn't specify any additional constraints, so perhaps we can set some variables to zero to find the others.Alternatively, maybe the assistant's suggestion implies that the new complexity function should maintain the same complexity for the classical composition when z is added, but that's already given as 250.Wait, perhaps the assistant is suggesting that the new complexity function should be an extension of the old one, meaning that when z = 0, the complexity should reduce to the old function. But in this case, z = 7, so that might not help.Alternatively, perhaps the new constants d, e, f, g are to be determined such that the complexity increases from 150 to 250 when z is added. But without more information, it's difficult to find unique values.Wait, perhaps the assistant's function is a quadratic in three variables, and we need to express it in terms of the new constants. But with only one equation, we can't solve for four variables.Wait, maybe the problem expects us to set some of the new constants to zero. For example, perhaps e = f = 0, and solve for d and g.But the problem doesn't specify that. Alternatively, perhaps the assistant's function is an extension, so the new terms involving z should account for the increase in complexity from 150 to 250.So, the increase is 100, which comes from the terms involving z: dz^2 + exz + fyz + iz + j. But since h = i = j = 0, the terms are dz^2 + exz + fyz.Given z = 7, x = 8, y = 5, the contribution from z terms is:d*(7)^2 + e*(8)(7) + f*(5)(7) = 49d + 56e + 35fSo, the increase in complexity is 100 = 49d + 56e + 35fBut in our earlier equation, we have:49d + 56e + 35f + 8g = 3215/23 ≈ 139.7826So, 49d + 56e + 35f = 100Therefore,8g = 3215/23 - 100 ≈ 139.7826 - 100 = 39.7826So,g = (3215/23 - 100)/8Compute 3215/23:3215 ÷ 23 = 139.7826139.7826 - 100 = 39.782639.7826 / 8 ≈ 4.9728But let's compute exactly:3215/23 - 100 = (3215 - 2300)/23 = 915/23So,8g = 915/23Thus,g = 915/(23*8) = 915/184 ≈ 4.9728So, g = 915/184Now, we have:49d + 56e + 35f = 100We need to find d, e, f such that this holds. Again, with three variables and one equation, we can't find unique solutions. So, perhaps we can set some variables to zero.For simplicity, let's set e = 0 and f = 0, then solve for d:49d = 100 => d = 100/49 ≈ 2.0408But then, e = f = 0, g = 915/184 ≈ 4.9728Alternatively, we can set d = 0, then:56e + 35f = 100We can set e = 0, then f = 100/35 ≈ 2.8571Or set f = 0, then e = 100/56 ≈ 1.7857But without additional constraints, we can't determine unique values for d, e, f.Wait, perhaps the problem expects us to set e = f = 0, as in the first sub-problem, and only introduce d and g. So, let's proceed with that.Set e = 0, f = 0, then:49d = 100 => d = 100/49And g = 915/184So, the new constants would be:d = 100/49 ≈ 2.0408e = 0f = 0g = 915/184 ≈ 4.9728But let's check if this makes sense.Plugging back into the equation:49*(100/49) + 56*0 + 35*0 + 8*(915/184) = 100 + 0 + 0 + (8*915)/184Simplify:100 + (7320)/184 = 100 + 39.7826 ≈ 139.7826Which is 3215/23 ≈ 139.7826, so it checks out.Therefore, one possible solution is:d = 100/49, e = 0, f = 0, g = 915/184But perhaps the problem expects us to set e and f to zero, as in the first sub-problem, to maintain simplicity.Alternatively, maybe the assistant's function is designed such that the new terms involving z contribute to the complexity without affecting the existing terms. But without more information, it's hard to say.Alternatively, perhaps the assistant's function is supposed to maintain the same complexity when z = 0, but in this case, z = 7, so that's not directly applicable.Given the lack of additional constraints, I think the most straightforward solution is to set e = f = 0, solve for d and g as above.So, final values:d = 100/49e = 0f = 0g = 915/184But let me express these as fractions:100/49 is already a fraction.915/184 can be simplified:Divide numerator and denominator by GCD(915,184). Let's compute GCD(915,184):915 ÷ 184 = 4 with remainder 179184 ÷ 179 = 1 with remainder 5179 ÷ 5 = 35 with remainder 45 ÷ 4 = 1 with remainder 14 ÷ 1 = 4 with remainder 0So, GCD is 1. Therefore, 915/184 is in simplest terms.So, the new constants are:d = 100/49e = 0f = 0g = 915/184But let me check if this makes sense in the context of the problem.When z = 0, the complexity should reduce to the original function, which was 150. Let's check:C(8,5,0) = a*64 + b*40 + c*25 + 0 + 0 + 0 + g*8 + 0 + 0 + 0= (15/23)*64 + (30/23)*40 + (15/23)*25 + (915/184)*8Compute each term:(15/23)*64 = 960/23 ≈ 41.739(30/23)*40 = 1200/23 ≈ 52.174(15/23)*25 = 375/23 ≈ 16.296(915/184)*8 = (915*8)/184 = 7320/184 = 39.7826Sum these:41.739 + 52.174 + 16.296 + 39.7826 ≈ 150Which is correct, as 960/23 + 1200/23 + 375/23 + 7320/184 = (960 + 1200 + 375)/23 + 7320/184 = 2535/23 + 7320/184Convert 2535/23 to 184 denominator:2535/23 = (2535*8)/184 = 20280/1847320/184 is already in denominator 184.So, total = (20280 + 7320)/184 = 27600/184 = 150Yes, correct.Therefore, the solution is consistent.So, summarizing sub-problem 2:d = 100/49e = 0f = 0g = 915/184h = 0i = 0j = 0But the problem only asks for d, e, f, g, so those are the values.</think>

🧠:<think>Okay, so I have this problem about a mask that's designed using parametric equations, and I need to calculate its surface area. Then, there's a transformation matrix applied to it, and I have to figure out how that affects the surface area. Hmm, let me try to break this down step by step.First, the surface S is given by the parametric equations:x(u, v) = cos(u) sin(v)y(u, v) = sin(u) sin(v)z(u, v) = cos(v)where u is between 0 and 2π, and v is between 0 and π. That looks familiar—it seems like the parametric equations for a sphere. Let me check: if I square x, y, and z and add them up, I should get 1.x² + y² + z² = cos²(u) sin²(v) + sin²(u) sin²(v) + cos²(v)= sin²(v)(cos²(u) + sin²(u)) + cos²(v)= sin²(v)(1) + cos²(v)= sin²(v) + cos²(v)= 1Yep, that's a unit sphere. So, the surface area of a unit sphere is 4π. But wait, the problem is asking me to calculate it using the parametric equations, so I can't just rely on that formula. I need to compute the surface integral.To find the surface area, I remember that the formula is the double integral over the parameters u and v of the magnitude of the cross product of the partial derivatives of the parametric equations. So, first, I need to find the partial derivatives of x, y, z with respect to u and v.Let me compute the partial derivatives:For x(u, v):∂x/∂u = -sin(u) sin(v)∂x/∂v = cos(u) cos(v)For y(u, v):∂y/∂u = cos(u) sin(v)∂y/∂v = sin(u) cos(v)For z(u, v):∂z/∂u = 0∂z/∂v = -sin(v)So, now I have the partial derivatives. The next step is to find the cross product of the two tangent vectors, which are ∂r/∂u and ∂r/∂v, where r is the position vector (x, y, z).Let me denote:∂r/∂u = ( -sin(u) sin(v), cos(u) sin(v), 0 )∂r/∂v = ( cos(u) cos(v), sin(u) cos(v), -sin(v) )Now, the cross product ∂r/∂u × ∂r/∂v is given by the determinant of the following matrix:|i j k||-sin(u)sin(v) cos(u)sin(v) 0||cos(u)cos(v) sin(u)cos(v) -sin(v)|Calculating the determinant:i * [cos(u)sin(v)*(-sin(v)) - 0*sin(u)cos(v)] - j * [-sin(u)sin(v)*(-sin(v)) - 0*cos(u)cos(v)] + k * [-sin(u)sin(v)*sin(u)cos(v) - cos(u)sin(v)*cos(u)cos(v)]Let me compute each component:i component:cos(u)sin(v)*(-sin(v)) - 0 = -cos(u) sin²(v)j component:- [ (-sin(u)sin(v))*(-sin(v)) - 0 ] = - [ sin(u) sin²(v) ] = -sin(u) sin²(v)Wait, hold on, the cross product formula is:If we have vectors (a1, a2, a3) and (b1, b2, b3), then the cross product is:(a2b3 - a3b2, a3b1 - a1b3, a1b2 - a2b1)So, maybe it's better to compute it that way.So, for ∂r/∂u × ∂r/∂v:First component (i): (∂y/∂u * ∂z/∂v - ∂z/∂u * ∂y/∂v)= [cos(u) sin(v) * (-sin(v)) - 0 * sin(u) cos(v)]= -cos(u) sin²(v)Second component (j): (∂z/∂u * ∂x/∂v - ∂x/∂u * ∂z/∂v)= [0 * cos(u) cos(v) - (-sin(u) sin(v)) * (-sin(v))]= 0 - sin(u) sin²(v)= -sin(u) sin²(v)Third component (k): (∂x/∂u * ∂y/∂v - ∂y/∂u * ∂x/∂v)= [ -sin(u) sin(v) * sin(u) cos(v) - cos(u) sin(v) * cos(u) cos(v) ]= [ -sin²(u) sin(v) cos(v) - cos²(u) sin(v) cos(v) ]= -sin(v) cos(v) [ sin²(u) + cos²(u) ]= -sin(v) cos(v) [1]= -sin(v) cos(v)So, the cross product vector is:( -cos(u) sin²(v), -sin(u) sin²(v), -sin(v) cos(v) )Now, to find the magnitude of this vector:|∂r/∂u × ∂r/∂v| = sqrt[ (-cos(u) sin²(v))² + (-sin(u) sin²(v))² + (-sin(v) cos(v))² ]Let me compute each term:First term: cos²(u) sin⁴(v)Second term: sin²(u) sin⁴(v)Third term: sin²(v) cos²(v)So, adding them up:cos²(u) sin⁴(v) + sin²(u) sin⁴(v) + sin²(v) cos²(v)Factor sin²(v):sin²(v) [ cos²(u) sin²(v) + sin²(u) sin²(v) + cos²(v) ]Wait, actually, let me factor sin²(v) from the first two terms:sin²(v) [ cos²(u) sin²(v) + sin²(u) sin²(v) ] + sin²(v) cos²(v)= sin²(v) [ sin²(v)(cos²(u) + sin²(u)) ] + sin²(v) cos²(v)= sin²(v) [ sin²(v)(1) ] + sin²(v) cos²(v)= sin⁴(v) + sin²(v) cos²(v)Factor sin²(v):= sin²(v)(sin²(v) + cos²(v)) = sin²(v)(1) = sin²(v)So, the magnitude is sqrt(sin²(v)) = |sin(v)|. Since v is between 0 and π, sin(v) is non-negative, so |sin(v)| = sin(v).Therefore, the surface area integral becomes:∫ (v=0 to π) ∫ (u=0 to 2π) sin(v) du dvCompute the inner integral first with respect to u:∫ (u=0 to 2π) sin(v) du = sin(v) * (2π - 0) = 2π sin(v)Then, integrate with respect to v:∫ (v=0 to π) 2π sin(v) dv = 2π [ -cos(v) ] from 0 to π = 2π [ -cos(π) + cos(0) ] = 2π [ -(-1) + 1 ] = 2π [1 + 1] = 4πSo, the surface area is 4π, which matches the known surface area of a unit sphere. That makes sense because the parametric equations given are indeed for a unit sphere.Okay, that was part 1. Now, moving on to part 2.The artist uses a transformation matrix T to distort the mask. The matrix is:T = [1 0.5 0] [0 1 0] [0 0 2]So, this is a linear transformation matrix. To apply this transformation to the parametric equations, I need to multiply the matrix T with the position vector (x, y, z).Wait, actually, the transformation is applied as T multiplied by the vector (x, y, z). So, the new coordinates (x', y', z') will be:x' = 1*x + 0.5*y + 0*z = x + 0.5 yy' = 0*x + 1*y + 0*z = yz' = 0*x + 0*y + 2*z = 2 zSo, substituting the original parametric equations:x'(u, v) = x + 0.5 y = cos(u) sin(v) + 0.5 sin(u) sin(v)y'(u, v) = y = sin(u) sin(v)z'(u, v) = 2 z = 2 cos(v)So, simplifying x'(u, v):x'(u, v) = sin(v) [ cos(u) + 0.5 sin(u) ]Hmm, that's the new parametric equations.Now, the question is about the effect of this transformation on the surface area. Would it increase, decrease, or remain the same?I remember that when you apply a linear transformation to a surface, the surface area scales by the determinant of the transformation matrix. But wait, actually, it's not exactly the determinant, because the determinant gives the scaling factor for volume, not area. For area, it's related to the square root of the determinant of (J^T J), where J is the Jacobian matrix of the transformation.But in this case, since the transformation is linear, the scaling factor for area is the square root of the determinant of (T^T T). Alternatively, the area scaling factor is the product of the singular values of T, but since T is diagonal, the singular values are just the absolute values of the diagonal entries.Wait, actually, for a linear transformation, the area scaling factor is the product of the scaling factors in each principal direction. But since T is a diagonal matrix except for the (1,2) entry, it's not just a simple scaling; it also has a shear component.Wait, let me think again. The transformation matrix T is:[1 0.5 0][0 1 0][0 0 2]So, it's a combination of a shear in the x-y plane and a scaling in the z-direction.To find the effect on surface area, I can consider the differential area element. The original surface has a differential area dA = |∂r/∂u × ∂r/∂v| du dv. After the transformation, the new differential area dA' is |∂r'/∂u × ∂r'/∂v| du dv.So, I need to compute the cross product of the transformed partial derivatives and find its magnitude.Alternatively, since the transformation is linear, the scaling factor for the area can be found by the square root of the determinant of (T^T T). Let me recall that for a linear transformation, the area scaling factor is sqrt(det(T^T T)).Wait, actually, for a linear transformation, the area scaling factor is the square root of the determinant of (J^T J), where J is the Jacobian matrix of the transformation. Since T is the Jacobian here (because it's a linear transformation), then the area scaling factor is sqrt(det(T^T T)).Let me compute T^T T:T^T is:[1 0 0][0.5 1 0][0 0 2]Multiplying T^T and T:First row of T^T times first column of T: 1*1 + 0*0 + 0*0 = 1First row of T^T times second column of T: 1*0.5 + 0*1 + 0*0 = 0.5First row of T^T times third column of T: 1*0 + 0*0 + 0*2 = 0Second row of T^T times first column of T: 0.5*1 + 1*0 + 0*0 = 0.5Second row of T^T times second column of T: 0.5*0.5 + 1*1 + 0*0 = 0.25 + 1 = 1.25Second row of T^T times third column of T: 0.5*0 + 1*0 + 0*2 = 0Third row of T^T times first column of T: 0*1 + 0*0 + 2*0 = 0Third row of T^T times second column of T: 0*0.5 + 0*1 + 2*0 = 0Third row of T^T times third column of T: 0*0 + 0*0 + 2*2 = 4So, T^T T is:[1 0.5 0 ][0.5 1.25 0 ][0 0 4 ]Now, the determinant of this matrix is:det(T^T T) = 1*(1.25*4 - 0*0) - 0.5*(0.5*4 - 0*0) + 0*(0.5*0 - 1.25*0)= 1*(5) - 0.5*(2) + 0= 5 - 1 + 0= 4So, sqrt(det(T^T T)) = sqrt(4) = 2.Therefore, the area scaling factor is 2. So, the surface area after transformation is 2 times the original surface area.Wait, but let me verify this because sometimes I get confused between volume and area scaling.Alternatively, another way to compute the area scaling factor is to compute the norm of the cross product of the transformed tangent vectors.Let me try that approach.First, the original tangent vectors are ∂r/∂u and ∂r/∂v.After transformation, the new tangent vectors are T*(∂r/∂u) and T*(∂r/∂v).So, let's compute these.First, ∂r/∂u = ( -sin(u) sin(v), cos(u) sin(v), 0 )Applying T:x' component: 1*(-sin(u) sin(v)) + 0.5*(cos(u) sin(v)) + 0*0 = -sin(u) sin(v) + 0.5 cos(u) sin(v)y' component: 0*(-sin(u) sin(v)) + 1*(cos(u) sin(v)) + 0*0 = cos(u) sin(v)z' component: 0*(-sin(u) sin(v)) + 0*(cos(u) sin(v)) + 2*0 = 0So, T*(∂r/∂u) = ( -sin(u) sin(v) + 0.5 cos(u) sin(v), cos(u) sin(v), 0 )Similarly, ∂r/∂v = ( cos(u) cos(v), sin(u) cos(v), -sin(v) )Applying T:x' component: 1*(cos(u) cos(v)) + 0.5*(sin(u) cos(v)) + 0*(-sin(v)) = cos(u) cos(v) + 0.5 sin(u) cos(v)y' component: 0*(cos(u) cos(v)) + 1*(sin(u) cos(v)) + 0*(-sin(v)) = sin(u) cos(v)z' component: 0*(cos(u) cos(v)) + 0*(sin(u) cos(v)) + 2*(-sin(v)) = -2 sin(v)So, T*(∂r/∂v) = ( cos(u) cos(v) + 0.5 sin(u) cos(v), sin(u) cos(v), -2 sin(v) )Now, compute the cross product of T*(∂r/∂u) and T*(∂r/∂v):Let me denote:A = T*(∂r/∂u) = ( a1, a2, a3 ) = ( -sin(u) sin(v) + 0.5 cos(u) sin(v), cos(u) sin(v), 0 )B = T*(∂r/∂v) = ( b1, b2, b3 ) = ( cos(u) cos(v) + 0.5 sin(u) cos(v), sin(u) cos(v), -2 sin(v) )The cross product A × B is:|i j k||a1 a2 a3||b1 b2 b3|So, compute each component:i component: a2*b3 - a3*b2 = cos(u) sin(v)*(-2 sin(v)) - 0*sin(u) cos(v) = -2 cos(u) sin²(v)j component: a3*b1 - a1*b3 = 0*(cos(u) cos(v) + 0.5 sin(u) cos(v)) - [ -sin(u) sin(v) + 0.5 cos(u) sin(v) ]*(-2 sin(v)) Let me compute this step by step:First term: 0*(...) = 0Second term: - [ (-sin(u) sin(v) + 0.5 cos(u) sin(v)) * (-2 sin(v)) ]= - [ (sin(u) sin(v) - 0.5 cos(u) sin(v)) * 2 sin(v) ]= - [ 2 sin(u) sin²(v) - cos(u) sin²(v) ]= -2 sin(u) sin²(v) + cos(u) sin²(v)So, the j component is 0 -2 sin(u) sin²(v) + cos(u) sin²(v) = (-2 sin(u) + cos(u)) sin²(v)Wait, actually, I think I made a mistake in the sign. Let me re-examine:The j component is a3*b1 - a1*b3.a3 is 0, so the first term is 0.The second term is -a1*b3.a1 = -sin(u) sin(v) + 0.5 cos(u) sin(v)b3 = -2 sin(v)So, -a1*b3 = - [ (-sin(u) sin(v) + 0.5 cos(u) sin(v)) * (-2 sin(v)) ]= - [ (sin(u) sin(v) - 0.5 cos(u) sin(v)) * 2 sin(v) ]= - [ 2 sin(u) sin²(v) - cos(u) sin²(v) ]= -2 sin(u) sin²(v) + cos(u) sin²(v)So, the j component is (-2 sin(u) + cos(u)) sin²(v)Wait, no, actually, it's (-2 sin(u) sin²(v) + cos(u) sin²(v)) which can be factored as sin²(v)(-2 sin(u) + cos(u)).But let's keep it as is for now.Third component (k): a1*b2 - a2*b1= [ (-sin(u) sin(v) + 0.5 cos(u) sin(v)) * sin(u) cos(v) ] - [ cos(u) sin(v) * (cos(u) cos(v) + 0.5 sin(u) cos(v)) ]Let me compute each part:First part: (-sin(u) sin(v) + 0.5 cos(u) sin(v)) * sin(u) cos(v)= [ -sin²(u) sin(v) cos(v) + 0.5 sin(u) cos(u) sin(v) cos(v) ]Second part: cos(u) sin(v) * (cos(u) cos(v) + 0.5 sin(u) cos(v))= cos²(u) sin(v) cos(v) + 0.5 sin(u) cos(u) sin(v) cos(v)So, subtracting the second part from the first part:[ -sin²(u) sin(v) cos(v) + 0.5 sin(u) cos(u) sin(v) cos(v) ] - [ cos²(u) sin(v) cos(v) + 0.5 sin(u) cos(u) sin(v) cos(v) ]= -sin²(u) sin(v) cos(v) + 0.5 sin(u) cos(u) sin(v) cos(v) - cos²(u) sin(v) cos(v) - 0.5 sin(u) cos(u) sin(v) cos(v)Simplify:The 0.5 sin(u) cos(u) sin(v) cos(v) terms cancel out.= -sin²(u) sin(v) cos(v) - cos²(u) sin(v) cos(v)Factor out -sin(v) cos(v):= -sin(v) cos(v) (sin²(u) + cos²(u)) = -sin(v) cos(v) (1) = -sin(v) cos(v)So, the k component is -sin(v) cos(v)Therefore, the cross product A × B is:( -2 cos(u) sin²(v), (-2 sin(u) + cos(u)) sin²(v), -sin(v) cos(v) )Now, let's compute the magnitude of this vector:|A × B| = sqrt[ (-2 cos(u) sin²(v))² + [ (-2 sin(u) + cos(u)) sin²(v) ]² + (-sin(v) cos(v))² ]Compute each term:First term: 4 cos²(u) sin⁴(v)Second term: [ (-2 sin(u) + cos(u))² sin⁴(v) ] = [4 sin²(u) - 4 sin(u) cos(u) + cos²(u)] sin⁴(v)Third term: sin²(v) cos²(v)So, adding them up:4 cos²(u) sin⁴(v) + [4 sin²(u) - 4 sin(u) cos(u) + cos²(u)] sin⁴(v) + sin²(v) cos²(v)Let me factor sin⁴(v) from the first two terms:sin⁴(v) [4 cos²(u) + 4 sin²(u) - 4 sin(u) cos(u) + cos²(u)] + sin²(v) cos²(v)Simplify inside the brackets:4 cos²(u) + 4 sin²(u) + cos²(u) - 4 sin(u) cos(u)= (4 cos²(u) + cos²(u)) + 4 sin²(u) - 4 sin(u) cos(u)= 5 cos²(u) + 4 sin²(u) - 4 sin(u) cos(u)Hmm, that seems complicated. Maybe there's a better way to approach this.Wait, perhaps instead of computing the cross product directly, I can use the fact that the area scaling factor is sqrt(det(T^T T)) = 2, as I found earlier. So, the surface area should scale by 2, making the new surface area 8π.But let me see if that's consistent with the cross product approach.Wait, the original cross product magnitude was sin(v), and now the new cross product magnitude is sqrt[4 cos²(u) sin⁴(v) + (4 sin²(u) - 4 sin(u) cos(u) + cos²(u)) sin⁴(v) + sin²(v) cos²(v)]But this seems messy. Maybe I can factor sin²(v) out of all terms:= sin²(v) [4 cos²(u) sin²(v) + (4 sin²(u) - 4 sin(u) cos(u) + cos²(u)) sin²(v) + cos²(v)]Wait, no, that's not correct because the third term is sin²(v) cos²(v), which is sin²(v) * cos²(v). So, actually, it's:= sqrt[ sin²(v) [4 cos²(u) sin²(v) + (4 sin²(u) - 4 sin(u) cos(u) + cos²(u)) sin²(v) + cos²(v) ] ]Wait, that still seems complicated. Maybe I made a mistake in the cross product calculation.Alternatively, perhaps it's better to accept that the area scaling factor is 2, so the new surface area is 8π. That would make sense because the determinant of T^T T is 4, so sqrt(4) = 2. Therefore, the surface area doubles.But let me think again: the original surface area is 4π, so after transformation, it should be 8π. That seems logical because the transformation stretches the sphere in the z-direction by a factor of 2 and shears it in the x-y plane. The shear doesn't change the area scaling because shear transformations preserve area in the plane, but the scaling in z would stretch the surface, increasing the area.Wait, but shear does affect the area. No, actually, shear transformations do change the area. For example, shearing a rectangle into a parallelogram changes the area if the shear is not along the axis. Wait, no, actually, shear transformations preserve area. Because the determinant of a shear matrix is 1, so the area scaling factor is 1. But in this case, the shear is combined with scaling.Wait, the determinant of T is 1*1*2 = 2. But determinant relates to volume scaling, not area. So, the area scaling factor is sqrt(det(T^T T)) = 2, as I found earlier.Therefore, the surface area after transformation is 2 times the original, so 8π.So, the surface area increases.Wait, but let me verify this with the cross product approach. If I compute the magnitude of the cross product, it should be 2 sin(v), because the original was sin(v), and the scaling factor is 2.But when I computed the cross product, I got:sqrt[4 cos²(u) sin⁴(v) + (4 sin²(u) - 4 sin(u) cos(u) + cos²(u)) sin⁴(v) + sin²(v) cos²(v)]Let me factor sin²(v) from all terms:= sqrt[ sin²(v) [4 cos²(u) sin²(v) + (4 sin²(u) - 4 sin(u) cos(u) + cos²(u)) sin²(v) + cos²(v) ] ]Wait, no, that's not correct because the third term is sin²(v) cos²(v), which is sin²(v) * cos²(v). So, actually, it's:= sqrt[ sin²(v) [4 cos²(u) sin²(v) + (4 sin²(u) - 4 sin(u) cos(u) + cos²(u)) sin²(v) + cos²(v) ] ]But that seems too complicated. Maybe I made a mistake in the cross product calculation.Alternatively, perhaps I can use the fact that the cross product magnitude is |T*(∂r/∂u) × T*(∂r/∂v)| = |T| * |∂r/∂u × ∂r/∂v|, but that's not quite accurate because the cross product involves the transformation of the vectors.Wait, actually, the cross product transforms as T*(∂r/∂u) × T*(∂r/∂v) = det(T) (∂r/∂u × ∂r/∂v) if T is orthogonal, but T is not orthogonal here.Wait, no, that's not correct. The cross product transformation is more complex. The magnitude of the cross product after transformation is equal to the determinant of T times the original cross product magnitude if T is orthogonal, but T is not orthogonal here.Wait, perhaps I should use the formula for the area scaling factor, which is sqrt(det(T^T T)) = 2, so the new cross product magnitude is 2 sin(v), which would make the integral 2 sin(v) over the same limits, leading to 2*4π = 8π.Alternatively, let me compute the integral with the new cross product magnitude.Wait, the original integral was ∫∫ sin(v) du dv = 4π.If the new cross product magnitude is 2 sin(v), then the new integral would be ∫∫ 2 sin(v) du dv = 2*4π = 8π.So, that seems consistent.Therefore, the surface area increases by a factor of 2, so the new surface area is 8π.So, to answer the question: Would the surface area increase, decrease, or remain the same after the transformation? It would increase. The mathematical justification is that the transformation matrix T has a scaling factor on area of sqrt(det(T^T T)) = 2, so the surface area doubles.</think>

🧠:<think>Alright, so I've got this problem about voting probabilities for three political parties: A, B, and C. It's divided into two parts. Let me take it step by step.Starting with part 1: The ratio of votes between Party A and Party B is 3:2, and Party C receives half the votes that Party A does. I need to find the probabilities ( p_A ), ( p_B ), and ( p_C ).Hmm, ratios can sometimes be tricky, but I think I can handle this. If the ratio of A to B is 3:2, that means for every 3 votes A gets, B gets 2. So, I can represent their probabilities as multiples of some common variable. Let's say the common variable is x. So, ( p_A = 3x ) and ( p_B = 2x ).Now, Party C gets half the votes that Party A does. So, ( p_C = frac{1}{2} p_A ). Since ( p_A = 3x ), then ( p_C = frac{1}{2} times 3x = frac{3}{2}x ).But wait, all the probabilities should add up to 1. So, ( p_A + p_B + p_C = 1 ). Plugging in the expressions:( 3x + 2x + frac{3}{2}x = 1 )Let me compute that:First, combine the terms:3x + 2x is 5x, and then adding ( frac{3}{2}x ) gives ( 5x + 1.5x = 6.5x ).So, ( 6.5x = 1 ). To find x, I can divide both sides by 6.5:( x = frac{1}{6.5} )But 6.5 is the same as ( frac{13}{2} ), so ( x = frac{2}{13} ).Now, let's find each probability:( p_A = 3x = 3 times frac{2}{13} = frac{6}{13} )( p_B = 2x = 2 times frac{2}{13} = frac{4}{13} )( p_C = frac{3}{2}x = frac{3}{2} times frac{2}{13} = frac{3}{13} )Let me check if they add up to 1:( frac{6}{13} + frac{4}{13} + frac{3}{13} = frac{13}{13} = 1 ). Perfect, that works.So, part 1 is done. The probabilities are ( p_A = frac{6}{13} ), ( p_B = frac{4}{13} ), and ( p_C = frac{3}{13} ).Moving on to part 2: There's a major political event that shifts voter behavior. The probability of voting for Party A increases by 20%, and the probability for Party B decreases by 10%. I need to recalculate the probabilities ( p_A' ), ( p_B' ), and ( p_C' ) ensuring they still sum to 1, and analyze the impact on Party C's probability.Alright, so first, let's figure out what a 20% increase and a 10% decrease mean in terms of the original probabilities.Starting with ( p_A ): It was ( frac{6}{13} ). A 20% increase would be ( p_A' = p_A + 0.2p_A = 1.2p_A ).Similarly, for ( p_B ): It was ( frac{4}{13} ). A 10% decrease would be ( p_B' = p_B - 0.1p_B = 0.9p_B ).So, let's compute these:( p_A' = 1.2 times frac{6}{13} = frac{7.2}{13} )( p_B' = 0.9 times frac{4}{13} = frac{3.6}{13} )Now, what about ( p_C' )? Since the total probabilities must still add up to 1, we can find ( p_C' ) by subtracting ( p_A' + p_B' ) from 1.So, ( p_C' = 1 - p_A' - p_B' )Let me compute ( p_A' + p_B' ):( frac{7.2}{13} + frac{3.6}{13} = frac{10.8}{13} )Therefore, ( p_C' = 1 - frac{10.8}{13} = frac{13}{13} - frac{10.8}{13} = frac{2.2}{13} )Wait, let me write that as a fraction. 2.2 is 11/5, so:( p_C' = frac{11}{5 times 13} = frac{11}{65} )But let me double-check that:Alternatively, 2.2 divided by 13 is 0.1692 approximately, but 11/65 is approximately 0.1692 as well. So, that seems correct.But let me represent all the new probabilities as fractions:( p_A' = frac{7.2}{13} = frac{72}{130} = frac{36}{65} )( p_B' = frac{3.6}{13} = frac{36}{130} = frac{18}{65} )( p_C' = frac{2.2}{13} = frac{22}{130} = frac{11}{65} )So, in simplified fractions:( p_A' = frac{36}{65} ), ( p_B' = frac{18}{65} ), ( p_C' = frac{11}{65} )Let me verify that they add up to 1:( 36 + 18 + 11 = 65 ), so yes, ( frac{65}{65} = 1 ). Perfect.Now, analyzing the impact on Party C's probability. Originally, ( p_C = frac{3}{13} approx 0.2308 ) or 23.08%. After the shift, ( p_C' = frac{11}{65} approx 0.1692 ) or 16.92%. So, Party C's probability has decreased by approximately 6.16 percentage points.Wait, let me compute the exact decrease:Original ( p_C = frac{3}{13} approx 0.23077 )New ( p_C' = frac{11}{65} approx 0.16923 )Difference: ( 0.23077 - 0.16923 = 0.06154 ), which is about 6.154%.So, Party C's probability has decreased by roughly 6.15%.But let me think about why that happened. When Party A's probability increased and Party B's decreased, the total shift from A and B affects Party C. The increase in A and decrease in B might have caused a reallocation of probabilities from C to A and B, but in this case, the net effect is that C lost some probability.Wait, actually, when A increased by 20%, that's an absolute increase, and B decreased by 10%, which is an absolute decrease. So, the total change is an increase of ( 0.2 times frac{6}{13} = frac{1.2}{13} ) for A, and a decrease of ( 0.1 times frac{4}{13} = frac{0.4}{13} ) for B. So, the net change is an increase of ( frac{1.2}{13} - frac{0.4}{13} = frac{0.8}{13} ). Therefore, the total probability has increased by ( frac{0.8}{13} ), which means Party C must decrease by that amount to keep the total at 1.So, the decrease in Party C's probability is exactly ( frac{0.8}{13} = frac{8}{130} = frac{4}{65} approx 0.0615 ), which is about 6.15%, matching our earlier calculation.Therefore, Party C's probability decreased by ( frac{4}{65} ) or approximately 6.15%.So, summarizing the changes:- Party A increased from ( frac{6}{13} ) to ( frac{36}{65} )- Party B decreased from ( frac{4}{13} ) to ( frac{18}{65} )- Party C decreased from ( frac{3}{13} ) to ( frac{11}{65} )All right, I think that covers both parts of the problem.</think>

🧠:<think>Okay, so I have this problem to solve, and it's about matrix multiplication and determinants. I'm a bit nervous because I haven't worked with matrices in a while, but let me try to break it down step by step.First, the problem says that there's an algorithm which multiplies two matrices A and B, resulting in matrix C. We need to verify that the determinant of C is equal to the product of the determinants of A and B. The matrices given are both 2x2, which is nice because calculating determinants for 2x2 matrices is straightforward.Let me write down the matrices:Matrix A:[2, 1][3, 4]Matrix B:[0, 5][6, 7]So, determinant of A, det(A), is calculated as (2*4) - (1*3) = 8 - 3 = 5.Similarly, determinant of B, det(B), is (0*7) - (5*6) = 0 - 30 = -30.Now, if the property holds, det(C) should be det(A) * det(B) = 5 * (-30) = -150.But wait, I should also compute C = A * B and then find det(C) to verify this.Let me compute matrix C.To multiply A and B:First row of A times first column of B: (2*0) + (1*6) = 0 + 6 = 6First row of A times second column of B: (2*5) + (1*7) = 10 + 7 = 17Second row of A times first column of B: (3*0) + (4*6) = 0 + 24 = 24Second row of A times second column of B: (3*5) + (4*7) = 15 + 28 = 43So matrix C is:[6, 17][24, 43]Now, determinant of C is (6*43) - (17*24) = 258 - 408 = -150.Yes, that matches det(A)*det(B) = -150. So the property holds here.Okay, that part makes sense. I think I did that correctly.Now, moving on to the second part. It's about the time complexity of matrix multiplication. The original algorithm has a time complexity of O(n^3), which is the standard for matrix multiplication. But they've implemented an optimization using Strassen's algorithm, which reduces the time complexity to O(n^{log2 7}).I remember that Strassen's algorithm is more efficient than the standard method, especially for large matrices. The exponent log2 7 is approximately 2.807, which is less than 3, so it should indeed be faster.The original algorithm takes 8 minutes to multiply two 256x256 matrices. We need to find the approximate time it would take using the optimized algorithm.Hmm, so let's think about how time complexity works. The time taken is proportional to n^k, where k is the exponent. So, if T1 is the time for the original algorithm and T2 is the time for the optimized one, then:T1 / T2 = (n^3) / (n^{log2 7}) ) = n^{3 - log2 7}We can solve for T2:T2 = T1 / (n^{3 - log2 7})But wait, actually, since the time is proportional to n^k, the ratio of times is (n1^k1)/(n2^k2) if the problem sizes are different. But in this case, the problem size is the same, n=256, so actually, the ratio of times is (n^3)/(n^{log2 7}) = n^{3 - log2 7}.So, T2 = T1 / (n^{3 - log2 7})But let me make sure about this. The original time is T1 = c1 * n^3, and the optimized time is T2 = c2 * n^{log2 7}. Assuming that the constants c1 and c2 are similar, which might not be the case, but for the sake of approximation, perhaps we can ignore the constants.But actually, Strassen's algorithm has a higher constant factor, so it's only faster for sufficiently large n. But since n=256 is quite large, maybe the constants are manageable.But the problem says "approximate time", so maybe they just want the ratio based on the exponents.So, let's compute the exponent first.Compute 3 - log2 7.We know that log2 7 is approximately 2.807, so 3 - 2.807 ≈ 0.193.So, the ratio is n^{0.193}.Given that n=256, which is 2^8, so 256^{0.193} = (2^8)^{0.193} = 2^{8*0.193} ≈ 2^{1.544}.Compute 2^1.544. 2^1 = 2, 2^1.5 ≈ 2.828, 2^1.544 is a bit more. Let's approximate it as around 2.9.So, the ratio is approximately 2.9.Therefore, T2 ≈ T1 / 2.9 ≈ 8 / 2.9 ≈ 2.758 minutes.Wait, but hold on. If T1 is proportional to n^3 and T2 is proportional to n^{log2 7}, then T2 = T1 * (n^{log2 7}) / (n^3) ) = T1 * n^{log2 7 - 3} = T1 * n^{-0.193}.Wait, that would be T2 = T1 * (1 / n^{0.193}), which is the same as T1 / n^{0.193}.But n=256, so n^{0.193} ≈ 2.9 as above.So, T2 ≈ 8 / 2.9 ≈ 2.758 minutes, which is approximately 2.76 minutes.But let me double-check my steps.1. Original time T1 = 8 minutes for n=256.2. Original time complexity: O(n^3).3. Optimized time complexity: O(n^{log2 7}) ≈ O(n^{2.807}).4. The ratio of times is T1 / T2 = (n^3) / (n^{2.807}) = n^{0.193}.5. So, T2 = T1 / n^{0.193}.6. Compute n^{0.193} where n=256.But 256 is 2^8, so 256^{0.193} = 2^{8*0.193} ≈ 2^{1.544} ≈ 2.9.So, T2 ≈ 8 / 2.9 ≈ 2.76 minutes.Alternatively, if we compute 256^{0.193} more accurately:Compute log2(256^{0.193}) = 0.193 * log2(256) = 0.193 * 8 = 1.544.So, 2^{1.544} ≈ e^{1.544 * ln2} ≈ e^{1.544 * 0.693} ≈ e^{1.071} ≈ 2.919.So, 256^{0.193} ≈ 2.919.Thus, T2 ≈ 8 / 2.919 ≈ 2.74 minutes.So, approximately 2.74 minutes.But let me think if this is the correct approach.Alternatively, sometimes when comparing algorithms, the time ratio is (T2 / T1) = (n^{log2 7}) / (n^3) = n^{log2 7 - 3}.But since log2 7 ≈ 2.807, so 2.807 - 3 = -0.193.So, T2 = T1 * (n^{-0.193}) = T1 / (n^{0.193}).Which is the same as before.So, yes, 8 / 2.919 ≈ 2.74 minutes.Alternatively, if we use natural logs, maybe?Wait, but the exponents are in terms of base 2, so probably better to stick with base 2.Alternatively, maybe the question expects us to use the ratio of the exponents directly.Wait, another approach: the time complexity is O(n^3) vs O(n^{log2 7}).So, the ratio of the times is (n^{log2 7}) / (n^3) = n^{log2 7 - 3}.But since log2 7 ≈ 2.807, so 2.807 - 3 ≈ -0.193.So, the optimized time is T1 * (n^{-0.193}) = T1 / (n^{0.193}).Which is the same as before.So, 8 / (256^{0.193}) ≈ 8 / 2.919 ≈ 2.74 minutes.Alternatively, maybe we can compute 256^{0.193} as e^{0.193 * ln256}.Compute ln256: ln(256) = ln(2^8) = 8 ln2 ≈ 8 * 0.693 ≈ 5.545.So, 0.193 * 5.545 ≈ 1.071.So, e^{1.071} ≈ 2.919, same as before.So, 8 / 2.919 ≈ 2.74.So, approximately 2.74 minutes.But let me check if I can compute 256^{0.193} more accurately.Compute 0.193 * log10(256).Wait, log10(256) = log10(2^8) = 8 log10(2) ≈ 8 * 0.3010 ≈ 2.408.So, 0.193 * 2.408 ≈ 0.465.So, 10^{0.465} ≈ 10^{0.465} ≈ 2.91.Same result.So, 256^{0.193} ≈ 2.91.Thus, T2 ≈ 8 / 2.91 ≈ 2.75 minutes.So, approximately 2.75 minutes.But the question says "approximate time", so maybe we can round it to about 2.75 or 2.8 minutes.Alternatively, if we use more precise values:log2(7) is approximately 2.807350555.So, 3 - log2(7) ≈ 0.192649445.Compute 256^{0.192649445}.Since 256 = 2^8, so 256^{0.192649445} = 2^{8 * 0.192649445} ≈ 2^{1.54119556}.Compute 2^{1.54119556}.We know that 2^1 = 2, 2^1.5 ≈ 2.8284, 2^1.54119556.Compute the difference: 1.54119556 - 1.5 = 0.04119556.So, 2^{0.04119556} ≈ e^{0.04119556 * ln2} ≈ e^{0.04119556 * 0.69314718056} ≈ e^{0.02857} ≈ 1.0289.So, 2^{1.54119556} ≈ 2.8284 * 1.0289 ≈ 2.908.Thus, 256^{0.192649445} ≈ 2.908.So, T2 ≈ 8 / 2.908 ≈ 2.751 minutes.So, approximately 2.75 minutes.Rounding to two decimal places, 2.75 minutes.But maybe the question expects it in minutes and seconds? 0.75 minutes is 45 seconds, so 2 minutes and 45 seconds. But the question says "approximate time", so either way is fine, but probably in decimal minutes.Alternatively, if we use the exact value:log2(7) = ln7 / ln2 ≈ 1.945910149 / 0.69314718056 ≈ 2.807350555.So, 3 - log2(7) ≈ 0.192649445.Compute 256^{0.192649445}.As above, it's approximately 2.908.So, 8 / 2.908 ≈ 2.751 minutes.So, approximately 2.75 minutes.Alternatively, if we use the exact exponent:The time ratio is (n^{log2 7}) / (n^3) = n^{log2 7 - 3} = n^{-0.192649445}.So, T2 = T1 * n^{-0.192649445} = 8 * (256)^{-0.192649445} ≈ 8 / 2.908 ≈ 2.751.So, yes, 2.75 minutes.But let me think if there's another way to approach this.Alternatively, since Strassen's algorithm has a better exponent, but the base is different. The standard matrix multiplication is O(n^3), and Strassen's is O(n^{log2 7}).But the actual time also depends on the constants involved. However, since the problem states that the original algorithm takes 8 minutes, and we are to compute the time using the optimized algorithm, assuming that the constants are similar, which might not be the case in reality, but for the sake of the problem, we can proceed with the exponent comparison.So, I think my calculation is correct.Therefore, the determinant part is straightforward, and the time complexity part requires understanding of how the exponents affect the time.So, to recap:1. Compute det(A) = 5, det(B) = -30, det(C) = -150, which is indeed 5*(-30) = -150. So the property holds.2. The optimized algorithm reduces the time from 8 minutes to approximately 2.75 minutes for 256x256 matrices.I think that's it. I hope I didn't make any calculation errors, but I double-checked the steps, so I think it's correct.Final Answer1. The determinant of ( C ) is (boxed{-150}), confirming the property.2. The approximate time using the optimized algorithm is (boxed{2.75}) minutes.</think>