Question

The circumference of the base of a $10m$ high conical tent is $44metres.$ Calculate the length of canvas used in making the tent if width of canvas is $2m.$ (Use $\pi =\frac{22}{7}$).

Answer 1

The total amount of canvas required would be equal to the curved surface area of the cone.

The formula of the curved surface area of a cone with base radius ‘r’ and slant height ‘l’ is given as

Curved Surface Area $=\pi rl$

It is given that the circumference of the base is $44m.$

So, $2\pi r=44m$

$\Rightarrow r=\frac{44}{2\pi }$

$\Rightarrow r=\frac{44\times 7}{2\times 22}=7m$

It is given that the vertical height of the cone is $h=10m.$

To find the slant height $‘l^{\prime }$ to be used in the formula for Curved Surface Area we use the following relation

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

$=\sqrt{7^2+{10}^2}$

$=\sqrt{49+100}$

$=\sqrt{149}$

$\therefore l=\sqrt{149}m$

Now, substituting the values of $r=7m$ and slant height, $l=\sqrt{149}m$ and using $\pi =\frac{22}{7}$ in the formula of C.S.A, we get

Curved Surface Area $=\pi rl=\frac{22}{7}\times 7\times \sqrt{149}$

$=22\sqrt{149}$

Hence the curved surface area of the given cone is $22\sqrt{149}{m}^2$

Now, the width of the canvas is $2m.$

Area of the canvas required $=$ (Width of the canvas)(Length of the canvas)

Therefore, Length of the canvas $=\frac{\text{Area of canvas}}{\text{Width of canvas}}$

$=\frac{22\sqrt{149}}{2}$ [$\because$ Area of canvas $=$ Curved Surface area of cone]

$=11\sqrt{149}=134.27m$

Hence the length of canvas required is $134.27m$