If the volume of a sphere inscribed in a cube is $\frac{\pi}{6}$ cubic inches, what is the number of cubic inches in the volume of the cube?
I need to find the relationship between the radius of the sphere and the side length of the cube. I know that the diameter of the sphere is equal to the side length of the cube, since the sphere is inscribed in the cube. So, if I call the radius of the sphere $r$ and the side length of the cube $s$, I have $2r = s$. I also know that the volume of the sphere is $\frac{4}{3}\pi r^3$ and the volume of the cube is $s^3$. The problem gives me the volume of the sphere as $\frac{\pi}{6}$, so I can plug that into the formula and get $\frac{4}{3}\pi r^3 = \frac{\pi}{6}$. I can simplify this by multiplying both sides by $\frac{3}{4}$ and dividing both sides by $\pi$, and get $r^3 = \frac{1}{8}$. Taking the cube root of both sides, I get $r = \frac{1}{2}$. Now, I can use the equation $2r = s$ to find the side length of the cube. I get $s = 2r = 2 \cdot \frac{1}{2} = 1$. Finally, I can use the formula for the volume of the cube and get $s^3 = 1^3 = 1$. 