A right circular cylidner and a cone have equal base radius and equal height. If their curved surface are in the ratio 8:5, then the radius of the base to the height are in the ratio :

Options

A. `2:3`
B. `4:3`
C. `3:4`
D. `3:2`

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the given information
We have a right circular cylinder and a cone with equal base radius \( R \) and equal height \( H \). The curved surface areas of the cylinder and cone are in the ratio \( 8:5 \).

### Step 2: Write the formulas for the curved surface areas
The curved surface area (CSA) of a cylinder is given by:
\[ \text{CSA}_{\text{cylinder}} = 2\pi R H \]
The curved surface area of a cone is given by:
\[ \text{CSA}_{\text{cone}} = \pi R L \]
where \( L \) is the slant height of the cone.

### Step 3: Set up the ratio of the curved surface areas
According to the problem, we have:
\[ \frac{\text{CSA}_{\text{cylinder}}}{\text{CSA}_{\text{cone}}} = \frac{8}{5} \]
Substituting the formulas, we get:
\[ \frac{2\pi R H}{\pi R L} = \frac{8}{5} \]

### Step 4: Simplify the equation
We can cancel \( \pi R \) from both sides (assuming \( R \neq 0 \)):
\[ \frac{2H}{L} = \frac{8}{5} \]

### Step 5: Cross-multiply to find a relationship between \( H \) and \( L \)
Cross-multiplying gives:
\[ 2H \cdot 5 = 8L \quad \Rightarrow \quad 10H = 8L \quad \Rightarrow \quad \frac{H}{L} = \frac{8}{10} = \frac{4}{5} \]

### Step 6: Relate the slant height \( L \) to the radius \( R \) and height \( H \)
The slant height \( L \) of the cone can be calculated using the Pythagorean theorem:
\[ L = \sqrt{H^2 + R^2} \]

### Step 7: Substitute \( L \) in the ratio
From \( \frac{H}{L} = \frac{4}{5} \), we can substitute \( L \):
\[ \frac{H}{\sqrt{H^2 + R^2}} = \frac{4}{5} \]

### Step 8: Cross-multiply and square both sides
Cross-multiplying gives:
\[ 5H = 4\sqrt{H^2 + R^2} \]
Squaring both sides results in:
\[ 25H^2 = 16(H^2 + R^2) \]
Expanding and rearranging gives:
\[ 25H^2 = 16H^2 + 16R^2 \quad \Rightarrow \quad 9H^2 = 16R^2 \]

### Step 9: Find the ratio of \( R \) to \( H \)
Dividing both sides by \( H^2 \) and rearranging gives:
\[ \frac{R^2}{H^2} = \frac{9}{16} \quad \Rightarrow \quad \frac{R}{H} = \frac{3}{4} \]

### Conclusion
Thus, the ratio of the radius of the base to the height is:
\[ \frac{R}{H} = \frac{3}{4} \]