Answer
Please see below.
Work Step by Step
As we know, the volume of a sphere with radius $r$ equals $\frac{4}{3}\pi r^3$.
Consider the function $V(r)=\frac{4}{3}\pi r^3$, which is obviously continuous on the closed interval $[5,8]$. Since $V(5) \approx 523.60$ and $V(8) \approx 2144.66$, there must exist a sphere with radius $r_0 \in [5,8]$ such that $V(r_0)=1500$, according to the Intermediate Value Theorem (Please note that $523.60<1500<2144.66$).
