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

We are asked to find the ratio of the radius of the base to the height when a right circular cylinder and a right circular cone have equal bases and heights, and their curved surface areas are in the ratio 8 : 5.
Step 1: Let the radius and height be
- Let the radius of the base = \(r\)
- Let the height of both the cylinder and the cone = \(h\)
Step 2: Write formulas for curved surface areas
- Curved Surface Area (CSA) of cylinder: \( \text{CSA}_{cyl} = 2 \pi r h \)
- Curved Surface Area (CSA) of cone: \( \text{CSA}_{cone} = \pi r l \), where \(l\) is the slant height of the cone
- Slant height of cone: \( l = \sqrt{r^2 + h^2} \)
Step 3: Use given ratio of CSAs
- CSA ratio: \( \text{CSA}_{cyl} : \text{CSA}_{cone} = 8 : 5 \)
\[ \frac{2 \pi r h}{\pi r l} = \frac{8}{5} \implies \frac{2 h}{l} = \frac{8}{5} \implies l = \frac{5}{4} h \]
Step 4: Express slant height in terms of r and h
- Slant height: \( l = \sqrt{r^2 + h^2} \)
\[ \sqrt{r^2 + h^2} = \frac{5}{4} h \]
\[ r^2 + h^2 = \left(\frac{5}{4} h\right)^2 = \frac{25}{16} h^2 \]
\[ r^2 = \frac{25}{16} h^2 - h^2 = \frac{9}{16} h^2 \]
\[ r = \frac{3}{4} h \]
Step 5: Find the ratio
\[ r : h = \frac{3}{4} : 1 = 3 : 4 \]
Answer:
Radius : Height = 3 : 4