The base of a cone and a cylinder have the same radius 6 cm. They have also the same height 8 cm. The ratio of the curved surfaces of the cylinder to that of the cone is

Options

A. `4:3`
B. `5:3`
C. `8:5`
D. `8:3`

Answer

To find the ratio of the curved surface areas of a cylinder and a cone with the same radius and height, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the given values:**
- Radius (r) = 6 cm
- Height (h) = 8 cm

2. **Calculate the curved surface area of the cylinder:**
- The formula for the curved surface area of a cylinder is given by:
\[ \text{Curved Surface Area of Cylinder} = 2\pi rh \]
- Substitute the values:
\[ = 2 \times \pi \times 6 \times 8 \]
- Calculate:
\[ = 2 \times 6 \times 8 \times \pi = 96\pi \text{ cm}^2 \]

3. **Calculate the slant height of the cone:**
- The formula for the slant height (l) of a cone is:
\[ l = \sqrt{r^2 + h^2} \]
- Substitute the values:
\[ = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10 \text{ cm} \]

4. **Calculate the curved surface area of the cone:**
- The formula for the curved surface area of a cone is:
\[ \text{Curved Surface Area of Cone} = \pi r l \]
- Substitute the values:
\[ = \pi \times 6 \times 10 \]
- Calculate:
\[ = 60\pi \text{ cm}^2 \]

5. **Find the ratio of the curved surface areas:**
- The ratio of the curved surface area of the cylinder to that of the cone is:
\[ \text{Ratio} = \frac{\text{Curved Surface Area of Cylinder}}{\text{Curved Surface Area of Cone}} = \frac{96\pi}{60\pi} \]
- Simplifying the ratio:
\[ = \frac{96}{60} = \frac{8}{5} \]

6. **Final Result:**
- The ratio of the curved surface area of the cylinder to that of the cone is:
\[ 8 : 5 \]