$\text{How many metres of cloth will be required to make a conical tent, the radius of whose base is}21\text{metres and height is}28\text{metres. The width of the cloth is}5\text{metres. (Where}\pi =\frac{22}{7}\text{)}$

The correct option is 462.

To find the total length of cloth required to make the conical tent, we need to calculate the curved surface area (CSA) of the cone and then divide it by the width of the cloth.

Step 1: Identify the given values
Radius of the conical base, $r=21\text{metres}$
Height of the conical tent, $h=28\text{metres}$
Width of the cloth, $w=5\text{metres}$

Step 2: Calculate the slant height (l) of the tent
The formula for the slant height $l$ of a cone is:

$l=\sqrt{r^2+h^2}$

Substitute the given values of $r$ and $h$:

$l=\sqrt{{21}^2+{28}^2}$

$l=\sqrt{441+784}$

$l=\sqrt{1225}=35\text{metres}$

Step 3: Calculate the Curved Surface Area (CSA) of the conical tent
The formula for the curved surface area of a cone is:

$\text{CSA}=\pi rl$

Substitute $\pi =\frac{22}{7}$ , $r=21$ , and $l=35$ into the formula:

$\text{CSA}=\frac{22}{7}\times 21\times 35$

Simplifying the calculation:

$\text{CSA}=22\times 3\times 35$

$\text{CSA}=2310{\text{m}}^2$

Step 4: Determine the length of cloth required
The total area of the cloth must equal the curved surface area of the tent. Since the cloth is rectangular in shape, its area is given by:

$\text{Area of cloth}=\text{Length}\times \text{Width}$

Therefore, the length of the cloth is:

$\text{Length of cloth}=\frac{\text{Curved Surface Area}}{\text{Width of cloth}}$

$\text{Length of cloth}=\frac{2310}{5}=462\text{metres}$

Thus, 462 metres of cloth will be required to make the conical tent.