Question

A joker’s cap is in the form of a right circular cone of base radius 7cm and height 27cm. Find the area of the sheet required to make 10 such caps.

Answer 1

Hint: Area of the sheet required is equal to the surface area of the cap. As this is a cap, we have to take curved surface area.

Formula: $\text{curved surface area of cone=}\pi \text{rl}$, where, r = radius of cone and l = slant length of cone

Complete step-by-step answer:

r = radius of the cone
h = height of the cone
l = slant length of cone

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

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

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

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

$\begin{align}

  & =\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$=10\times \left( 613.023\ c{{m}^{2}} \right)$

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

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 $=\pi rl+\pi {{r}^{2}}$