G-6 Archimedes Formula

let $r$ be the radius of the sphere, then the height of the cylinder is $2r$

the volume of the sphere is given by the formula: $V_s=\dfrac{4}{3} \pi r^3$

the volume of the cylinder is given by the formula: $V_c=\pi r^2h$ where $h$ is the height. it follows that $V_c=\pi r^2(2r)=2 \pi r^3$

Finally, the ratio of their volume is

$ratio=\dfrac{\frac{4}{3}}{2}=\dfrac{4}{3} \times \dfrac{1}{2}=\dfrac{4}{6}=\dfrac{2}{3}$ or $\color{#D61F06}\boxed{\large 2:3}$

The main thing to know while solving this question is that the height of this cylinder equals to the diameter of Sphere.

The volume of the sphere is $\frac{4}{3}\pi r^{3}$

And the volume of the cylinder is $2\pi r^{3}$ since height equals to 2r

Then the ratio of the sphere to the cylinder is ( $\frac{4}{3}\pi r^{3}$ ) : ( $2\pi r^{3}$ ) = 2:3

Hence the answer is $\boxed{2:3}$

$\text{ratio}=\dfrac{\text{volume of sphere}}{\text{volume of cylinder}}=\dfrac{\frac{4}{3}\pi (r^3)}{\pi (r^2)(2r)}=\dfrac{\frac{4}{3}}{2}=\dfrac{2}{3}=\boxed{2:3}$

$\text{ratio of volumes}=\dfrac{\dfrac{4}{3} \pi r^3}{\pi r^2 \times 2r}=\dfrac{\dfrac{4}{3} \pi r^3}{2 \pi r^3}=\large{\boxed{\dfrac{2}{3}}}$