Question

A joker’s cap is in the form of a cone of radius 7 cm and height 24 cm. Find the area of the cardboard required to make the cap.

Answer 1

Hint- Here, we will be using the formula for curved surface area of a cone. A suitable figure needs to be drawn in which the pink colour will be determining the portion where cardboard is used so that we can clearly see the area used to make the joker’s cap.

Given, radius of the cone-shaped joker’s cap $r = 7{\text{ cm}}$
Height of the cone-shaped joker’s cap $h = 24{\text{ cm}}$
Since, the area of the cardboard required to make the cone-shaped cap is the curved surface area of the cone with the dimensions as that of the given cone-shaped cap.
As we know that the curved surface area of the cone with radius as $r$ and height as $h$ is given by $\pi rl$ where slant height is $l = \sqrt {{r^2} + {h^2}} $
$ \Rightarrow $Area of the cardboard required to make the cap$ = $Curved surface area of the cap$ = \pi rl = \pi r\sqrt {{r^2} + {h^2}} = \dfrac{{22}}{7} \times 7 \times \sqrt {{7^2} + {{24}^2}} = 22 \times 25 = 550{\text{ c}}{{\text{m}}^2}$

Note- In these types of problems, we usually observe the figure carefully and find out the area which is responsible for making the cone. Clearly, the cap material is used to make the conical curved surface of the cap that’s why the value of curved surface is calculated.