Question

A conical tent of radius of 12 m and height 16 m is to be made, then the cost of canvas required at the rate of Rs 10 per $ {{m}^{2}} $ is:
(a) Rs 7445
(b) Rs 7543
(c) Rs 7550
(d) Rs 7500

Answer 1

Hint: We have given the radius and height of the conical tent is 12 m and 16 m respectively. Canvas should be put on the curved surface area of the conical tent so find the curved surface area of the conical tent by using the formula for curved surface area (C.S.A) of cone which is equal to $ C.S.A=\pi rl $ . In this formula, “r” represents the radius and “l” represents the slant height of the cone. After finding the curved surface area of the cone, multiply this area by Rs 10 to get the cost of the canvas.

Complete step-by-step answer:
We have given the radius and height of the conical tent as 12 m and 16 m respectively.
In the below diagram, we have shown a cone of radius and height as r and h respectively.
 

In the above figure, CG is the slant height of the cone. Slant height of the cone is calculated by using Pythagoras theorem because $ \Delta CAG $ is a right angled triangle and CG is the hypotenuse of this triangle.
 $ {{\left( CG \right)}^{2}}={{\left( CA \right)}^{2}}+{{\left( AG \right)}^{2}} $
Substituting CA as h and AG as r in the above equation we get,
 $ {{\left( CG \right)}^{2}}={{\left( h \right)}^{2}}+{{\left( r \right)}^{2}} $
Taking square root on both the sides we get,
 $ \begin{align}
  & \sqrt{{{\left( CG \right)}^{2}}}=\sqrt{{{\left( h \right)}^{2}}+{{\left( r \right)}^{2}}} \\
 & \Rightarrow CG=\sqrt{{{\left( h \right)}^{2}}+{{\left( r \right)}^{2}}} \\
\end{align} $
Substituting the value of h and r as 16 m and 12 m respectively we get,
 $ \begin{align}
  & CG=\sqrt{{{\left( 16 \right)}^{2}}+{{\left( 12 \right)}^{2}}} \\
 & \Rightarrow CG=\sqrt{256+144} \\
 & \Rightarrow CG=\sqrt{400} \\
 & \Rightarrow CG=20m \\
\end{align} $
We have found the slant height as 20 m.
Now, we know that curved surface area (C.S.A) of the cone is equal to:
 $ C.S.A=\pi rl $
In the above equation, r and l represents the radius and slant height of the cone.
Substituting r as 12 m and l as 20 m in the C.S.A formula we get,
 $ \begin{align}
  & C.S.A=\pi \left( 12 \right)\left( 20 \right) \\
 & \Rightarrow C.S.A=\left( \dfrac{22}{7} \right)\left( 12 \right)\left( 20 \right) \\
 & \Rightarrow C.S.A=754.29{{m}^{2}} \\
\end{align} $
The canvas is put on the curved surface area of the conical tent so the cost of canvas for $ 753.6{{m}^{2}} $ area at the rate of Rs 10 per $ {{m}^{2}} $ is calculated by multiplying $ 753.6{{m}^{2}} $ to Rs 10.
 $ \begin{align}
  & 754.29\left( 10 \right) \\
 & =\text{Rs}7542.9 \\
\end{align} $
Rounding off the above amount to the nearest integer we get Rs 7543.
Hence, the correct option is (b).

Note: In the above problem, if you substitute the value of $ \pi $ in the formula of curved surface area then you will get the following value of curved surface area:
 $ C.S.A=\pi \left( 12 \right)\left( 20 \right) $
Substituting the value of $ \pi $ as 3.14 in the above equation we get,
 $ \begin{align}
  & C.S.A=\left( 3.14 \right)\left( 12 \right)\left( 20 \right) \\
 & \Rightarrow C.S.A=Rs753.6 \\
\end{align} $
Multiplying the above curved surface area by 10 will give the cost of canvas.
 $ \begin{align}
  & Rs\left( 753.6 \right)\left( 10 \right) \\
 & =Rs7536 \\
\end{align} $
Now, matching the above answer with the options given in the question you will find that no option is matching. This means that if accidently you take the value of $ \pi $ as 3.14 then you won’t get the correct option so you have to check the curved surface area value by substituting the value of $ \pi $ as $ \dfrac{22}{7} $ and then see whether your answer is matching or not.