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
Given the relation: C = (3)/(2) sqrt(f^2 + 4P^2Q) We need to express f in terms of C, P, and Q. Step 1: Multiply both sides by 2. 2C = 3 sqrt(f^2 + 4P^2Q) Step 2: Divide both sides by 3. (2C)/(3) = sqrt(f^2 + 4P^2Q) Step 3: Square both sides to eliminate the square root. ((2C)/(3))^2 = f^2 + 4P^2Q (4C^2)/(9) = f^2 + 4P^2Q Step 4: Subtract 4P^2Q from both sides to isolate f^2. f^2 = (4C^2)/(9) - 4P^2Q Step 5: Combine the terms on the right side by finding a common denominator. f^2 = (4C^2)/(9) - (36P^2Q)/(9) f^2 = (4C^2 - 36P^2Q)/(9) Step 6: Take the square root of both sides to solve for f. f = ± sqrt((4C^2 - 36P^2Q)/(9)) Step 7: Simplify the expression by taking the square root of the denominator and factoring out 4 from the numerator inside the square root. f = ± sqrt(4(C^2 - 9P^2Q))sqrt(9) f = ± 2sqrt(C^2 - 9P^2Q)3 The final answer is f = ± 2sqrt(C^2 - 9P^2Q)3.