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
Step 1: Write the variation equation. The problem states that M varies jointly as x^2y and inversely as Z. This can be written as: M = k (x^2y)/(Z) where k is the constant of proportionality. Step 2: Find the constant of proportionality (k) using the initial given values. We are given m=64 when x=2, y=4, and Z=1. Substitute these values into the equation: 64 = k ((2)^2(4))/(1) 64 = k (4 × 4)/(1) 64 = k × 16 To find k, divide both sides by 16: k = (64)/(16) k = 4 Step 3: Write the complete variation equation. Now that we have the value of k, the specific relationship between M, x, y, and Z is: M = 4 (x^2y)/(Z) Step 4: Find x using the new set of values. We need to find x when m=144, y=9, and Z=3. Substitute these values into the complete variation equation: 144 = 4 (x^2(9))/(3) Step 5: Solve for x. Simplify the right side of the equation: 144 = 4 × (3x^2) 144 = 12x^2 Divide both sides by 12: x^2 = (144)/(12) x^2 = 12 Take the square root of both sides to find x: x = sqrt(12) Simplify the square root: x = sqrt(4 × 3) x = 2sqrt(3) The value of x is 2sqrt(3).