Question

A joker’s cap in the form of a right circular cone whose base radius is 7 cm and height is 24 cm. Find the area of the sheet required to make 10 such caps.

Answer 1

Hint: Area of the sheet required to make the joker’s cap is equal to the surface area of the cap. As this is a cap, the base will be open and we have to take only curved surface area. Calculate curved surface area of the cap using formula $''CSA=\pi rl''$where r = radius of base and l = slant length and then multiply the CSA of one cap by 10 to get the area of the sheet required for making 10 such caps.


Complete step-by-step answer:

r = radius of the cone

h = height of the cone

l = slant length of the cone

We, know ${{h}^{2}}+{{r}^{2}}={{l}^{2}}$ (For a right circular cone)

As we have to find the area of the sheet making cap, the base will be open. So, we have to
consider only curved surface areas.

Curved surface area (CSA) $=\pi rl$

Given, base radius = r = 7cm

Height = h = 27cm

Using, ${{l}^{2}}={{h}^{2}}+{{r}^{2}}$

Taking square root both sides,

$\begin{align}

  & \Rightarrow l=\sqrt{{{h}^{2}}+{{r}^{2}}} \\

 & =\sqrt{{{\left( 27 \right)}^{2}}+{{\left( 7 \right)}^{2}}} \\

 & =\sqrt{729+49} \\

 & =\sqrt{778} \\

 & =27.89cm \\

\end{align}$

Now,

$\begin{align}

  & CSA=\pi \times \left( 7cm \right)\left( 27.89cm \right) \\

 & =\left( 3.14\times 7\times 27.89 \right)c{{m}^{2}} \\

 & =613.023c{{m}^{2}} \\

\end{align}$

Curved surface area (CSA) of the given cone $=613.023c{{m}^{2}}$.

So, the area of sheet required for making one such cap will be $613.023c{{m}^{2}}$.

But we have to find the area of the sheet required for making 10 such caps.

So, the required area of sheet will be 10 times of the area of sheet required for making one such cap.

Thus, the required area of sheet $=10\times \left( 613.023c{{m}^{2}} \right)$

$=6130.23c{{m}^{2}}$


Note: In this question, a student can do step by taking the total surface area instead of the curved surface area of the cone. But as it is a cap, the base will be open, so we have to consider only curved surface area.

For a cone, total surface area = curved surface area + area of base.

Curved surface area $=\pi rl$

Area of base $=\pi {{r}^{2}}$

Total surface area $=\pi rl+\pi {{r}^{2}}$