Question

Work out the ratio of the volume of the sphere to the volume of the cylinder

Answer 1

so we know the radius of the sphere is 8, and its surface area is 4πr², therefore its surface area is 4π8², namely 256π.

we also know that the surface area of the cylinder is half that of the sphere, namely 256π/2 or 128π, let's use that surface area to get the height h.

$\bf \textit{total surface area of a cylinder}\\\\ SA=2\pi r(h+r)~~ \begin{cases} r=4\\ SA=128\pi \end{cases}\implies 128\pi =2\pi (4)(h+4) \\\\\\ 128\pi =8\pi(h+4)\implies \cfrac{128\pi }{8\pi }=h+4\implies 16=h+4\implies \boxed{12=h}$

since now we know h and r for the cylinder, and we also know r for the sphere, let's check their volume's ratio.

$\bf \begin{array}{|c|ll} \cline{1-1}\\ \textit{volume of a sphere}\\\\ V=\cfrac{4\pi r^3}{3}\\\\ \textit{volume of a cylinder}\\\\ V=\pi r^2 h\\\\ \cline{1-1} \end{array}~\hspace{3em}\cfrac{~~\frac{4\pi 8^3}{3}~~}{\pi (4)^2(12)}\implies \cfrac{~~\frac{2048\pi }{3}~~}{192\pi } \\\\\\ \cfrac{2048\pi }{3}\cdot \cfrac{1}{192\pi }\implies \cfrac{32}{9}$