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:

When you see a ratio resulting in \(h/\sqrt{r^2+h^2} = 4/5\), recognize the 3-4-5 triplet. Here \(h=4\) corresponds to the hypotenuse 5, meaning the other side \(r\) must be 3. Thus, \(r/h = 3/4\).

Answer 1

Step 1: Understanding the Concept:
Let \(r\) be the radius and \(h\) be the height for both solids. The curved surface area (CSA) of a cylinder is \(2\pi rh\) and the CSA of a cone is \(\pi rl\), where \(l = \sqrt{r^2 + h^2}\).
Step 2: Key Formula or Approach:
\[ \frac{\text{CSA of Cylinder}}{\text{CSA of Cone}} = \frac{8}{5} \]
Step 3: Detailed Explanation:
\[ \frac{2\pi rh}{\pi r \sqrt{r^2 + h^2}} = \frac{8}{5} \]
Simplifying by cancelling \(\pi r\):
\[ \frac{2h}{\sqrt{r^2 + h^2}} = \frac{8}{5} \implies \frac{h}{\sqrt{r^2 + h^2}} = \frac{4}{5} \]
Squaring both sides:
\[ \frac{h^2}{r^2 + h^2} = \frac{16}{25} \]
Cross-multiplying:
\[ 25h^2 = 16r^2 + 16h^2 \]
\[ 9h^2 = 16r^2 \]
Taking the ratio \(r/h\):
\[ \frac{r^2}{h^2} = \frac{9}{16} \]
Taking the square root:
\[ \frac{r}{h} = \frac{3}{4} \]
Step 4: Final Answer:
The ratio between the radius and height is \(3 : 4\).