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
Here's the solution to your variation problem. The problem states that a varies directly as b and as the square of c. This can be written as an equation: a = kbc^2 where k is the constant of variation. We are given that a=30 when b=0.5 and c=2. a) Find the value of k, the constant of variation. Step 1: Substitute the given values into the variation equation. 30 = k(0.5)(2)^2 Step 2: Simplify and solve for k. 30 = k(0.5)(4) 30 = k(2) k = (30)/(2) k = 15 The constant of variation is 15. b) Find the value of a when b=2 and c=3. Step 1: Use the constant of variation k=15 found in part (a) and the new values for b and c. a = 15(2)(3)^2 Step 2: Calculate the value of a. a = 15(2)(9) a = 30(9) a = 270 The value of a is 270. c) Find the values of c when a=300 and b=4. Step 1: Use the constant of variation k=15 and the given values for a and b. 300 = 15(4)c^2 Step 2: Simplify and solve for c^2. 300 = 60c^2 c^2 = (300)/(60) c^2 = 5 Step 3: Take the square root of both sides to find c. Remember to include both positive and negative roots. c = ±sqrt(5) The values of c are sqrt(5) or -sqrt(5). 3 done, 2 left today. You're making progress.