Question

A right circular cylinder and a right circular cone have equal bases and equal heights. If their curved surface areas are in the ratio 8 : 5, then find the ratio between the radius of their bases to their height.

Hint:

From $\frac{h}{l} = \frac{4}{5}$, we can see that the height and the slant height form a 3-4-5 right triangle with the radius.
If height is 4 units and hypotenuse (slant height) is 5 units, the base radius must be 3 units.
This immediately gives the ratio of radius to height as $3 : 4$.

Answer 1

Step 1: Turn the CSA ratio into a ratio of height to slant height.
CSA of cylinder $=2\pi rh$ and CSA of cone $=\pi rl$, so $\dfrac{2\pi rh}{\pi rl}=\dfrac85 \implies \dfrac{2h}{l}=\dfrac85 \implies \dfrac{h}{l}=\dfrac45$.
Step 2: Recognise the right triangle formed by r, h, and l.
Since $l=\sqrt{r^2+h^2}$, the numbers $r$, $h$, $l$ form a right triangle. With $h:l=4:5$, this matches the classic 3-4-5 triple exactly, so we can write $h=4k$ and $l=5k$ for some $k$, which forces $r=3k$ by Pythagoras.
Step 3: Read off the required ratio.
$r:h = 3k:4k = 3:4$.
\[ \boxed{3:4} \]