Question

A tent of height 77 dm is in the form of a right circular cylinder of diameter 36m and height 44d m surmounted by a right circular cone. Find the cost of the canvas at \[Rs.3.50\;per\;{m^2}\] (Use \[\pi = \dfrac{{22}}{7}\] ).

Answer 1

Hint: Tent is a combination of circular cylinder and a cone. We will find the surface area of both cylinder and the cone and then will add them up to find the cost of canvas at the rate of \[Rs.3.50\;per\;{m^2}\] .
Surface area of cylinder is \[SA = 2\pi rh\]
Surface area of the cone is \[SA = \pi r\sqrt {{h^2} + {r^2}} \]

Complete step-by-step answer:
Given
The total height of the tent \[H = 77\;dm = 7.7\;m\]
The height of cylinder \[{h_1} = 44\;dm = 4.4\;m\]
The height of cone \[{h_2} = 7.7 - 4.4 = 3.3\;m\]
Diameter of right circular cylinder \[d = 36\;m\]
Now since in a tent right circular cone is mounted over a right circular cylinder hence we can say the radius of both cylinder and cone will be same which is
 \[r = \dfrac{d}{2} = \dfrac{{36}}{2} = 18\;m\]
Now to find the cost of the canvas we need to find the overall surface area of the tent which is a combination of a circular cylinder and a cone, hence we can write the area of the canvas \[A = 2\pi r{h_1} + \pi r\left( {\sqrt {h_2^2 + {r^2}} } \right)\]
By substituting the values in the equation we get
 \[
  A = 2 \times \dfrac{{22}}{7} \times 18 \times 4.4 + \dfrac{{22}}{7} \times 18\left( {\sqrt {{{\left( {3.3} \right)}^2} + {{\left( {18} \right)}^2}} } \right) \\
   = 497.82 + 56.57\left( {\sqrt {10.89 + 324} } \right) \\
   = 497.82 + 56.57\left( {\sqrt {334.89} } \right) \\
   = 497.82 + 56.57 \times 18.3 \\
   = 497.82 + 1035.231 \\
   = 1533.051 \;
 \]
Hence we get the total surface area of the tent \[ = 1533.051\;{m^2}\]
Now since to make a tent the given cost of canvas is \[Rs.3.50\;per\;{m^2}\] , hence the total cost to make a tent of surface area \[1533.051\;{m^2}\] will be \[ = 1533.051 \times 3.50 = Rs.5365.678\]
Therefore, the cost of the canvas at \[@Rs.3.50\;per\;{m^2}\] \[ = Rs.5366\]
So, the correct answer is “Rs.5366”.

Note: Surface area of a solid object is the measure of the total area that the surface of an object occupies and in this question since we needed to find the cost of canvas so we used the formula to find the surface area of the tent to find the total canvas required for the tent from which cost was calculated.