This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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You're on a roll — here's the solution for question 24: The relationship between A, B, and C is given by: A varies directly as B and inversely as the square of C. This can be written as: A = k (B)/(C^2) where k is the constant of proportionality. c) Write down an equation connecting A, B and C. Given A = 125 when B = 180 and C = 6. Step 1: Substitute the given values into the variation equation to find k. 125 = k (180)/(6^2) 125 = k (180)/(36) 125 = k × 5 Step 2: Solve for k. k = (125)/(5) k = 25 Step 3: Write the equation connecting A, B, and C using the value of k. A = (25B)/(C^2) d) Find the value of C when B = 96 and A = 150. Step 1: Use the equation found in part c). A = (25B)/(C^2) Step 2: Substitute A = 150 and B = 96 into the equation. 150 = (25 × 96)/(C^2) Step 3: Solve for C^2. 150 C^2 = 25 × 96 150 C^2 = 2400 C^2 = (2400)/(150) C^2 = 16 Step 4: Solve for C. C = sqrt(16) C = 4 e) Find the percentage increase in A if B increases by 10% and C increases by 15%. Let the original values be A_1, B_1, C_1. A_1 = (25B_1)/(C_1^2) Step 1: Calculate the new values of B and C. If B increases by 10%, the new B, B_2, is B_1 × (1 + 0.10) = 1.1 B_1. If C increases by 15%, the new C, C_2, is C_1 × (1 + 0.15) = 1.15 C_1. Step 2: Calculate the new value of A, A_2. A_2 = (25B_2)/(C_2^2) A_2 = (25(1.1 B_1))/((1.15 C_1)^2) A_2 = (25 × 1.1 B_1)/(1.15^2 C_1^2) A_2 = (1.1)/(1.15^2) × (25 B_1)/(C_1^2) Step 3: Substitute A_1 = (25B_1)/(C_1^2) into the equation for A_2. A_2 = (1.1)/(1.15^2) A_1 A_2 = (1.1)/(1.3225) A_1 A_2 ≈ 0.83175879 A_1 Step 4: Calculate the percentage increase. Percentage increase = ((A_2 - A_1)/(A_1)) × 100\% Percentage increase = ((A_2)/(A_1) - 1) × 100\% Percentage increase = ((1.1)/(1.3225) - 1) × 100\% Percentage increase = (0.83175879 - 1) × 100\% Percentage increase = -0.16824121 × 100\% Percentage increase ≈ -16.82\% The negative sign indicates a decrease. Percentage increase = -16.82\% What's next?