Question

A self-group wants to manufacture joker caps (conical shapes) of 3 cm radius and 4 cm height. If the available colour paper sheets are $ 1000\,c{{m}^{2}} $ , then how many caps can be manufactured from that paper sheet?

Answer 1

Hint: In this question, we have to first find out the lateral surface area of one joker cap by using the formula, LSA $ =\pi rl $ but in order to find this, you need to find the slant height which is $ l=\sqrt{{{r}^{2}}+{{h}^{2}}} $ . Now you need to divide the available colour sheets by the lateral surface area of one joker cap.

Complete step-by-step answer:
We have to manufacture joker caps with a radius of base 3 cm and 4 cm of height. We have paper sheets of about $ 1000\,c{{m}^{2}} $ with what we need to make the joker caps. So, let us calculate how many joker caps can be manufactured.

We know,
Lateral surface area of the cone = $ \pi rl $ , where r = radius of the cone = 3 cm and slant height $ l $ .
Let us find the value of slant height, $ l=\sqrt{{{r}^{2}}+{{h}^{2}}} $ , where r = 3 cm and height $ h=4\,cm $
  $ \begin{align}
  & l=\sqrt{{{r}^{2}}+{{h}^{2}}} \\
 & =\sqrt{{{3}^{2}}+{{4}^{2}}} \\
 & =\sqrt{9+16} \\
 & =\sqrt{25} \\
 & =5
\end{align} $
Now, we have the slant height as 5 cm.
Substitute the value of slant height, radius in the lateral surface area of the cone


We get,
Lateral surface area = $ \pi rl $
                                     $ \begin{align}
  & =\pi \left( 3 \right)\left( 5 \right) \\
 & =47.12 \\
\end{align} $
Here, we found out the lateral surface area is $ 47.12\,c{{m}^{2}} $
We also have the colour sheets of about $ 1000\,c{{m}^{2}} $
Now, let us divide the total number of sheets of $ 1000\,c{{m}^{2}} $ by the lateral surface area of one joker cap to find the total number of joker caps that can be manufactured.
Total number of joker caps to be manufactured = $ \dfrac{1000}{47.12} $
Number of caps = 21.22
We can say approximately 21 joker caps.
Hence, 21 joker caps can be manufactured by the available colour sheets.

Note: You need to keep in mind that to input the units of each and required measurement given. Also if you have been asked for volume of the cone its formula is $ V=\dfrac{1}{3}\pi {{r}^{2}}h $ and it’s total surface area is $ S{{A}_{total}}=\pi {{r}^{2}}+\pi rl $ .