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: Understanding the Question:
We have a right circular cylinder and a right circular cone.
They share:
- The same base radius $r$.
- The same vertical height $h$.
The ratio of their Curved Surface Areas (CSA) is given as $8 : 5$.
We need to find the ratio of their base radius to their height ($r : h$).

Step 2: Key Formula or Approach:
1. Curved Surface Area of a Cylinder $= 2\pi r h$.
2. Curved Surface Area of a Cone $= \pi r l$, where $l$ is the slant height given by:
\[ l = \sqrt{r^2 + h^2} \]
3. Express the ratio of their CSAs as an algebraic equation, simplify, and solve for $\frac{r}{h}$.

Step 3: Detailed Explanation:

• Set up the ratio of the CSAs:
\[ \frac{\text{CSA of Cylinder}}{\text{CSA of Cone}} = \frac{2\pi r h}{\pi r l} = \frac{8}{5} \]

• Cancel common terms ($\pi$ and $r$) from the numerator and denominator:
\[ \frac{2h}{l} = \frac{8}{5} \]
Divide both sides by 2:
\[ \frac{h}{l} = \frac{4}{5} \]

• Substitute the formula for slant height $l = \sqrt{r^2 + h^2}$:
\[ \frac{h}{\sqrt{r^2 + h^2}} = \frac{4}{5} \]

• Square both sides of the equation to eliminate the radical:
\[ \frac{h^2}{r^2 + h^2} = \frac{16}{25} \]

• Cross-multiply to solve the algebraic equation:
\[ 25h^2 = 16(r^2 + h^2) \]
\[ 25h^2 = 16r^2 + 16h^2 \]

• Group the $h^2$ terms on one side:
\[ 25h^2 - 16h^2 = 16r^2 \]
\[ 9h^2 = 16r^2 \]

• Rearrange to find the ratio $\frac{r^2}{h^2}$:
\[ \frac{r^2}{h^2} = \frac{9}{16} \]

• Take the square root of both sides to get the ratio of radius to height:
\[ \frac{r}{h} = \sqrt{\frac{9}{16}} = \frac{3}{4} \]


Step 4: Final Answer:
The ratio of the radius of their bases to their height is $3 : 4$.