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.
ScanSolve AI Answer
You're on a roll — 24. Variation c) Given that A = 125 when B = 180 and C = 6, write down an equation connecting A, B and C. Step 1: Write the variation relationship. A varies directly as B: A B A varies inversely as the square of C: A (1)/(C^2) Combining these, we get: A (B)/(C^2) Introducing the constant of proportionality, k: A = (kB)/(C^2) Step 2: Use the given values to find the constant k. Given A = 125, B = 180, C = 6. 125 = (k × 180)/(6^2) 125 = (180k)/(36) 125 = 5k k = (125)/(5) k = 25 Step 3: Write the equation connecting A, B, and C. Substitute the value of k back into the variation equation: 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 the given values A = 150 and B = 96. 150 = (25 × 96)/(C^2) Step 3: Solve for C. 150 C^2 = 25 × 96 150 C^2 = 2400 C^2 = (2400)/(150) C^2 = 16 C = sqrt(16) C = 4 (Assuming C is a positive value in this context). e) Find the percentage increase in A if B increases by 10% and C increases by 15%. Step 1: Express the new values of B and C in terms of their original values. Let the original values be B_1 and C_1. If B increases by 10%, the new value B_2 is: B_2 = B_1 + 0.10 B_1 = 1.10 B_1 If C increases by 15%, the new value C_2 is: C_2 = C_1 + 0.15 C_1 = 1.15 C_1 Step 2: Express the new value of A, A_2, using the variation equation. The original equation is A_1 = (25B_1)/(C_1^2). The new value A_2 is: A_2 = (25B_2)/(C_2^2) Substitute B_2 and C_2: A_2 = (25(1.10 B_1))/((1.15 C_1)^2) A_2 = (25 × 1.10 B_1)/(1.15^2 C_1^2) A_2 = (1.10)/(1.15^2) × (25B_1)/(C_1^2) Since A_1 = (25B_1)/(C_1^2): A_2 = (1.10)/(1.15^2) A_1 A_2 = (1.10)/(1.3225) A_1 A_2 ≈ 0.83175886 A_1 Step 3: Calculate the percentage increase in A. Percentage increase = ((A_2 - A_1)/(A_1)) × 100\% = ((0.83175886 A_1 - A_1)/(A_1)) × 100\% = (0.83175886 - 1) × 100\% = -0.16824114 × 100\% = -16.82\% (to 2 decimal places) Since the value is negative, it represents a percentage decrease. The percentage increase is negative. -16.82\% What's next?