Scale Factor Between Spheres

Question

What is the radius of a sphere whose volume is twice larger than that of a sphere of radius r?

I believe that this problem has something to do with scaling factor.

Answer 1

The formula for the volume of a sphere with radius $r$ is $\frac{4}{3}\pi r^3$. Let $S$ be a sphere with radius $r$ and $S_1$ be a sphere with radius $r_1$ and twice the volume of $S$. Then, we have $2\cdot\frac{4}{3}\pi r^3=\frac{4}{3}\pi {r_1}^3$. Solving for $r_1$ in terms of $r$, we have $r_1=\sqrt[3]{2}r$.

In general, the dimensions of a solid whose volume is $n$ times the volume of another similar solid is $\sqrt[3]{n}$ times the dimensions of the other solid.