Volumes Practice Test 3

Concept:For a cylinder and cone with the same base radius $r$ and height $h$, compare curved surface areas using the cone slant height $l=\sqrt{r^2+h^2}$.Explanation:Cylinder curved surface area $=2\pi rh$.Cone curved surface area $=\pi rl=\pi r\sqrt{r^2+h^2}$.Given ratio: $\frac{2\pi rh}{\pi r\sqrt{r^2+h^2}}=\frac{8}{5}$.Cancel $\pi r$: $\frac{2h}{\sqrt{r^2+h^2}}=\frac{8}{5}$.So, $\frac{h}{\sqrt{r^2+h^2}}=\frac{4}{5}$.Squaring: $\frac{h^2}{r^2+h^2}=\frac{16}{25}$.Thus, $25h^2=16r^2+16h^2$.So, $9h^2=16r^2$.Hence, $\frac{r^2}{h^2}=\frac{9}{16}$.Taking positive square root: $\frac{r}{h}=\frac{3}{4}$.Therefore, radius : height $=3:4$.Answer:The required ratio is $3:4$, so the correct option is C.