How many meters of cloth 10 m wide will be required to make a conical tent with base radius of 14m and height 48 m?

Options

A. 110 m
B. 55 m
C. 77 m
D. 220 m

Answer

To solve the problem of how many meters of cloth 10 m wide will be required to make a conical tent with a base radius of 14 m and height of 48 m, we will follow these steps:

### Step 1: Calculate the Slant Height of the Cone
The slant height (l) of a cone can be calculated using the Pythagorean theorem. The formula is:
\[ l = \sqrt{h^2 + r^2} \]
where \( h \) is the height of the cone and \( r \) is the radius of the base.

Given:
- Height \( h = 48 \, m \)
- Radius \( r = 14 \, m \)

Now substituting the values:
\[ l = \sqrt{48^2 + 14^2} \]
\[ l = \sqrt{2304 + 196} \]
\[ l = \sqrt{2500} \]
\[ l = 50 \, m \]

### Step 2: Calculate the Surface Area of the Cone
The lateral surface area (A) of a cone can be calculated using the formula:
\[ A = \pi r l \]

Substituting the values we have:
\[ A = \pi \times 14 \times 50 \]
\[ A = 700\pi \, m^2 \]

Using \( \pi \approx 3.14 \):
\[ A \approx 700 \times 3.14 \]
\[ A \approx 2198 \, m^2 \]

### Step 3: Determine the Amount of Cloth Required
Since the cloth is 10 m wide, we need to calculate how many meters of cloth will be required to cover the surface area of the cone.

The area covered by 1 meter of cloth (10 m wide) is:
\[ \text{Area covered by 1 meter of cloth} = 10 \, m \times 1 \, m = 10 \, m^2 \]

Now, to find the total length of cloth required:
\[ \text{Length of cloth} = \frac{\text{Total Surface Area}}{\text{Area covered by 1 meter of cloth}} \]
\[ \text{Length of cloth} = \frac{2198 \, m^2}{10 \, m^2} \]
\[ \text{Length of cloth} = 219.8 \, m \]

### Final Answer
To make the conical tent, approximately **219.8 meters** of cloth will be required.

---