A sphere is inscribed in a cone with height $3$ and base radius $4$. What is the ratio of the volume of the sphere to the volume of the cone?
Consider the cross section of the cone and sphere. The diagram below shows the cross section when we cut the cone and sphere vertically.
[asy]
real h=3;
draw(Circle((0,0), 4));
draw((-sqrt(7),1.2)--(sqrt(7),1.2));
draw((sqrt(7),1.2)--(0,7*h/8));
draw((0,7*h/8)--(-sqrt(7),1.2));
draw((0,0)--(0,7*h/8));
label("$4$", (sqrt(7),1.2), E);
label("$4$", (-sqrt(7),1.2), W);
label("$3$", (0,7*h/8), N);
[/asy]
Let $x$ be half the length of a side of the equilateral triangle formed by the circle's cross section. Since the altitude of an equilateral triangle divides the base into two $30^\circ$-$60^\circ$-$90^\circ$ right triangles, $x = 4 \tan 60^\circ = 4 \sqrt{3}.$ Hence, the radius of the circle is $2x = 8 \sqrt{3},$ and the volume of the sphere is
\[\frac{4}{3} \pi (8 \sqrt{3})^3 = \frac{4 \pi \cdot 8^3 \cdot 3^2}{3^4} = 256 \pi.\]The volume of the cone is
\[\frac{1}{3} \cdot \pi \cdot 4^2 \cdot 3 = 16 \pi.\]Therefore, the ratio of the volume of the sphere to the volume of the cone is
\[\frac{256 \pi}{16 \pi} = \boxed{16}.\]
To find the ratio of the volume of the sphere to the volume of the cone, we need to calculate the volumes of both shapes.
The volume of a sphere is given by the formula $V_{\text{sphere}} = \frac{4}{3}\pi r^3$, where $r$ is the radius.
The radius of the sphere is equal to the radius of the cone's base, which is given as $4$.
Substituting the values into the formula, we get $V_{\text{sphere}} = \frac{4}{3}\pi (4)^3 = \frac{256}{3}\pi$.
The volume of a cone is given by the formula $V_{\text{cone}} = \frac{1}{3}\pi r^2 h$, where $r$ is the radius and $h$ is the height.
In this case, the radius of the cone's base is $4$, and the height is given as $3$.
Substituting the values into the formula, we get $V_{\text{cone}} = \frac{1}{3}\pi (4)^2 (3) = \frac{48}{3}\pi = 16\pi$.
So, the ratio of the volume of the sphere ($V_{\text{sphere}}$) to the volume of the cone ($V_{\text{cone}}$) is $\frac{V_{\text{sphere}}}{V_{\text{cone}}} = \frac{\frac{256}{3}\pi}{16\pi} = \frac{16}{3}$.