The surface area of the first ball is related to the surface area of the second ball as 5: 3.

The surface area of the first ball is related to the surface area of the second ball as 5: 3. Find the ratio of the volume of the first ball to the volume of the second ball.

Let us find the ratio of the radii of the balls, if the ratio of the area of the first ball to the area of the second is 5: 3. We denote the radius of the first ball r₁, the radius of the second ball r₂:
4πr₁²: 4πr₂² = 5: 3;
r₁: r₂ = √ (5/3);
r₁ = r₂ * √ (5/3).
Let’s find the volume of the first ball, expressing it in terms of the radius of the second:
4 / 3π (r₂ * √ (5/3) ³ = 4/3 * r₂³ * 5/3 * √ (5/3) = 20/9 * √ (5/3) * r₂³
Let’s find the volume of the second ball:
4 / 3πr₂³.
Let’s find the ratio of the volume of the first ball to the second:
20/9 * √ (5/3): 4/3 = 5/3 * √ (5/3)
Answer: the ratio of the volume of the first ball to the second is 5/3 * √ (5/3).