Question

a cone and a sphere have the same volume. the height of the cone is 96 units. what would be the values for the radius of the cone and the sphere?

Answer 1

The volume $V_c$ of a cone  with base radius $r_c$ and height $h_c$ is

$V_c = \dfrac{1}{3}\pi r_c^2 h_c$

Similarly, the volume $V_s$ of a sphere with radius $r_s$ is

$V_s = \dfrac{4}{3}\pi r_s^3$

We know that $V_c=V_s$ and that $h_c=96$

So, we can set up the following equation:

$\dfrac{96}{3}\pi r_c^2=\dfrac{4}{3}\pi r_s^3$

We can simplify the common denominator 3, and pi appearing on both sides:

$96r_c^2=4r_s^3$

We can divide both sides by 4:

$24r_c^2=r_s^3$

Without further information, this is all we can say: the cubed radius of the sphere is the same as 24 times the squared radius of the cone.