Express the radius of a sphere as a function of the sphere’s surface area. Then express the surface area as a function of the volume.

Verified step by step guidance

1. Start by recalling the formula for the surface area of a sphere, which is given by \( S = 4\pi r^2 \), where \( S \) is the surface area and \( r \) is the radius.
2. To express the radius \( r \) as a function of the surface area \( S \), solve the surface area formula for \( r \). Begin by dividing both sides by \( 4\pi \) to isolate \( r^2 \): \( r^2 = \frac{S}{4\pi} \).
3. Take the square root of both sides to solve for \( r \): \( r = \sqrt{\frac{S}{4\pi}} \). This expresses the radius as a function of the surface area.
4. Next, recall the formula for the volume of a sphere, which is \( V = \frac{4}{3}\pi r^3 \), where \( V \) is the volume.
5. To express the surface area \( S \) as a function of the volume \( V \), first solve the volume formula for \( r \): \( r = \left(\frac{3V}{4\pi}\right)^{1/3} \). Substitute this expression for \( r \) into the surface area formula \( S = 4\pi r^2 \) to express \( S \) in terms of \( V \).