This physics problem tests your understanding of fundamental physical laws and their applications. The step-by-step solution below breaks down the problem using relevant equations and physical reasoning.
ScanSolve AI Answer
Step 1: Establish the relationship between density, mass, and radius. The density of a solid spherical ball varies directly as its mass m and inversely as the cube of its radius r. This can be written as: = (k · m)/(r^3) where k is the constant of proportionality. Step 2: Calculate the constant of proportionality k. We are given the initial conditions: m = 500 g, r = 5 cm, and = 2 g/cm³. Substitute these values into the equation: 2 = (k · 500)/(5^3) 2 = (k · 500)/(125) 2 = 4k Divide by 4 to solve for k: k = (2)/(4) k = 0.5 Step 3: Calculate the radius for the new conditions. We need to find the radius r when the mass m = 340 g and the density = 10 g/cm³. Use the calculated value of k = 0.5: 10 = (0.5 · 340)/(r^3) 10 = (170)/(r^3) Multiply both sides by r^3 and divide by 10 to solve for r^3: r^3 = (170)/(10) r^3 = 17 Take the cube root of both sides to find r: r = [3]17 r ≈ 2.571 The radius of the solid spherical ball is approximately 2.57 cm. 2.57 cm That's 2 down. 3 left today — send the next one.