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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Here's the solution to question 3, parts i) and ii): From the previous question, the relationship between A, B, and C is: A = k (B^2)/(C^2) i) Given that when A=20, B=5 and C=9. Find A when B=7 and C=25. Step 1: Find the constant k using the initial values A=20, B=5, C=9. Substitute these values into the equation: 20 = k (5^2)/(9^2) 20 = k (25)/(81) To solve for k, multiply both sides by (81)/(25): k = 20 × (81)/(25) k = (4 × 5 × 81)/(5 × 5) k = (4 × 81)/(5) k = (324)/(5) k = 64.8 Step 2: Use the value of k to find A when B=7 and C=25. Substitute k=64.8, B=7, and C=25 into the equation: A = 64.8 (7^2)/(25^2) A = 64.8 (49)/(625) A = (64.8 × 49)/(625) A = (3175.2)/(625) A = 5.08032 Rounding to a reasonable number of decimal places, for example, two: A ≈ 5.08 The value of A when B=7 and C=25 is 5.08. ii) If B is increased by 20% and C decreased by 36%, find the percentage change in A. 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 is increased by 20%, the new value B_2 is: B_2 = B_1 + 0.20 B_1 = 1.20 B_1 If C is decreased by 36%, the new value C_2 is: C_2 = C_1 - 0.36 C_1 = 0.64 C_1 Step 2: Write the original and new expressions for A. Original A (A_1): A_1 = k (B_1^2)/(C_1^2) New A (A_2): A_2 = k (B_2^2)/(C_2^2) Substitute B_2 = 1.20 B_1 and C_2 = 0.64 C_1 into the equation for A_2: A_2 = k ((1.20 B_1)^2)/((0.64 C_1)^2) A_2 = k (1.44 B_1^2)/(0.4096 C_1^2) A_2 = (1.44)/(0.4096) ( k (B_1^2)/(C_1^2) ) Recognize that k (B_1^2)/(C_1^2) is A_1: A_2 = (1.44)/(0.4096) A_1 A_2 = 3.515625 A_1 Step 3: Calculate the percentage change in A. Percentage change = New Value - Original ValueOriginal Value × 100\% Percentage change in A = (A_2 - A_1)/(A_1) × 100\% Percentage change in A = (3.515625 A_1 - A_1)/(A_1) × 100\% Percentage change in A = (3.515625 - 1) × 100\% Percentage change in A = 2.515625 × 100\% Percentage change in A = 251.5625\% Rounding to two decimal places: Percentage change in A ≈ 251.56\% The percentage change in A is an increase of 251.56\%. 3 done, 2 left today. You're making progress.