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

Options

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

Answer

To solve the problem of how many metres of cloth will be required to make a conical tent with a base radius of 14 m and a height of 48 m, we need to find the surface area of the conical tent. The surface area of a cone is given by the formula:

\[ \text{Surface Area} = \pi r (r + l) \]

where:
- \( r \) is the radius of the base,
- \( l \) is the slant height of the cone.

### Step 1: Find the slant height \( l \)
To find the slant height \( l \), we can use the Pythagorean theorem. The slant height is the hypotenuse of a right triangle formed by the radius and the height of the cone.

The formula for the slant height \( l \) is:

\[ l = \sqrt{r^2 + h^2} \]

where:
- \( r = 14 \, m \) (base radius),
- \( h = 48 \, m \) (height).

Substituting the values:

\[ l = \sqrt{14^2 + 48^2} \]
\[ l = \sqrt{196 + 2304} \]
\[ l = \sqrt{2500} \]
\[ l = 50 \, m \]

### Step 2: Calculate the surface area
Now that we have the slant height, we can substitute the values into the surface area formula:

\[ \text{Surface Area} = \pi r (r + l) \]
\[ \text{Surface Area} = \pi \times 14 \times (14 + 50) \]
\[ \text{Surface Area} = \pi \times 14 \times 64 \]
\[ \text{Surface Area} = 896\pi \]

### Step 3: Calculate the amount of cloth required
Since the width of the cloth is 10 m, we need to find out how many metres of cloth will be required.

The length of cloth required can be calculated by dividing the surface area by the width of the cloth:

\[ \text{Length of cloth} = \frac{\text{Surface Area}}{\text{Width}} \]
\[ \text{Length of cloth} = \frac{896\pi}{10} \]
\[ \text{Length of cloth} = 89.6\pi \]

Using \( \pi \approx 3.14 \):

\[ \text{Length of cloth} \approx 89.6 \times 3.14 \]
\[ \text{Length of cloth} \approx 281.344 \, m \]

### Final Answer
Thus, the total length of cloth required is approximately **281.34 metres**.

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