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.
[tex] \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} [/tex]
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.
[tex] \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} [/tex]
