Question

What size does the radius of a sphere need to be for its volume to be larger than its surface area? HINT! It is less than 10. HINT! It is NOT a whole number. If you can show me a whole number for a radius where the surface area and volume are equal, then any radius bigger than that will have a larger volume.
To get full points you will need to SHOW me formulas for surface area of a sphere, volume of a sphere, and calculations on how you found your answer.
Question 1 options:
The formula for the surface area of a sphere is 4 x pi x radius^2
The formula for the volume of a sphere is 4/3 x pi x radius^3.

Answer 1

The volume will be larger than the surface area for any $r$ satisfying the inequality

$\dfrac43\pi r^3>4\pi r^2$

Since $r>0$ must be true, we can divide through by $4\pi r^2$ to get

$\dfrac13r>1\implies r>3$