Question

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

Answer 1

Hint: When we asked to calculate the sheet required for the making of the cone, we need to calculate the curved surface area of the cone. We know that the curve’s surface area of the cone is given by $ A=\pi rl $ where $ r $ is the base radius of the cone and $ l $ is the curved length of the cone. In the problem, we have the base radius only, instead of curved length or curved height we have the height of the cone and the cone is right circular. For a right circular cone, we have the relation between the height of the cone, the radius of the cone, curved length as $ {{l}^{2}}={{h}^{2}}+{{r}^{2}} $ , from this we can calculate the length of the curve of the cap which is in a cone shape. From this value, we can calculate the curved surface area of the cone which gives the sheet required to make one cap. To calculate the sheet required for the ten caps we will multiply the curved surface area of one cap with ten.

Complete step by step answer:
Given that, Joker’s cap is in the form of a right circular cone, which is shown below

Base radius $ r=7 $ cm
Height $ h=24 $ cm.
Now the inclined length can be calculated by
 $ \begin{align}
  & l=\sqrt{{{r}^{2}}+{{h}^{2}}} \\
 & \Rightarrow l=\sqrt{{{7}^{2}}+{{24}^{2}}} \\
 & \Rightarrow l=\sqrt{49+576} \\
 & \Rightarrow l=\sqrt{625} \\
 & \Rightarrow l=25cm \\
\end{align} $
Now we have the inclined or curved length as $ l=25 $ cm.
So, the value of curved surface area of the cap is equal to the curved surface area of the cone which is having base radius $ r=7 $ cm and curved length $ l=25 $ cm. Hence
 $ \begin{align}
  & A=\pi rl \\
 & \Rightarrow A=3.14\times 7\times 25 \\
 & \Rightarrow A=549.5 \\
 & \Rightarrow A\simeq 550c{{m}^{2}} \\
\end{align} $
So, the sheet required to manufacture one joker cap according to given data is $ 500c{{m}^{2}} $ . At the same time the sheet required to manufacture ten caps is the multiple of $ 10 $ to the sheet required for one cap.
 $ \therefore 10\times A=10\times 550=5500c{{m}^{2}} $
Hence the sheet required for the ten joker’s caps is $ 5500c{{m}^{2}} $ .

Note:
The problems related to geometrical shapes are very interesting when you only know what to find and what we have in the problem. In this problem, they have asked to calculate the sheet required. We know that sheet will cover the cone’s outer surface only. So, we have calculated the surface area of the cone. But when they have asked questions like how much amount of liquid or water can be filled in the cone, then we need to calculate the volume of the cone.