If a cone and sphere have equal radii and volumes, then determine the ratio of the diameter of the sphere to the height of the cone?

1:2

The correct answer is Option (2) → 1:2

Let the common radius be $r$.

Volume of the sphere is given by:

$\quad V_{\text{sphere}} = \frac{4}{3} \pi r^3$

Volume of the cone is given by:

$\quad V_{\text{cone}} = \frac{1}{3} \pi r^2 h$

Since the volumes are equal,

$\quad \frac{4}{3} \pi r^3 = \frac{1}{3} \pi r^2 h$

$\quad 4r = h$

So, height of the cone is:

$\quad h = 4r$

Now, diameter of the sphere is:

$\quad D = 2r$

Required ratio of diameter of the sphere to height of the cone is:

$\quad \frac{2r}{h} = \frac{2r}{4r} = \frac{1}{2}$