Question

A circus tent is made up of canvas and is in the form of a right circular cylinder and right circular cone above it. The diameter and height of the cylinder part are 126m and 5m respectively. The total height of the tent is 21m. Find the total cost of the tent if the canvas used costs Rs.36 per square meter.

Answer 1

Hint: To find the total cost of tent required for canvas for making the tent we need to find the curved surface area of the tent. To find the height of the cone we will remove the height of the cylinder from the total height of the tent. After finding the curved surface area of the cone and cylinder, we will add them to get the total curved surface area of the tent. And then to find the cost we will multiply the area by the cost of per square meter. This is the final answer.

Step by step solution:
Very first we will draw a well labeled diagram of the tent.

Now from the diagram we can easily get the dimensions. So let’s tabulate the areas.

Area of base of tent	Cylinder
Formula used	Curved surface area \[ = 2\pi rh\]
Dimensions	r= \[\dfrac{{126}}{2} = 63m\]h=5m
calculations	Curved surface area \[ = 2\pi rh\]Substituting the values in the formula,\[ \Rightarrow 2 \times \dfrac{{22}}{7} \times 63 \times 5\]Cancelling 63 by 7 we get,\[ \Rightarrow 2 \times 22 \times 9 \times 5\]On multiplying we get,\[ \Rightarrow 1980{m^2}\]

Now in order to find the area of the cone we have to find the height of the cone. Now we will remove the height of the cylinder from the total height of the tent.
Height of cone = \[21 - 5 = 16m\]
We have to find the slant height of the cone also.
 

We are given the radius and height of the cone. So using Pythagoras theorem we will calculate the slant height.
\[l = \sqrt {{h^2} + {r^2}} \]
Putting the values of h and r we get,
\[ \Rightarrow l = \sqrt {{{16}^2} + {{63}^2}} \]
Taking the squares,
\[ \Rightarrow l = \sqrt {256 + 3969} \]
On adding them we get,
\[ \Rightarrow l = \sqrt {4225} \]
Taking the root we get,
 \[ \Rightarrow l = 65m\]
This is the length of slant height.
Now let’s calculate the area.

Area of top of tent	Right circular cone
Formula used	Curved surface area = \[\pi rl\]
Dimensions	r=63ml=65m
calculations	Curved surface area = \[\pi rl\]Substituting the values we get,\[ \Rightarrow \dfrac{{22}}{7} \times 63 \times 65\]Divide 63 by 7 \[ \Rightarrow 22 \times 9 \times 65\]On multiplying we get,\[ \Rightarrow 12870{m^2}\]

Now the total curved surface area of the tent is the sum of both the curved surface areas.
\[CS{A_{tent}} = CS{A_{cylinder}} + CS{A_{cone}}\]
\[ \Rightarrow CS{A_{tent}} = 1980 + 12870\]
On adding we get,
\[ \Rightarrow CS{A_{tent}} = 14850{m^2}\]
This is our curved surface area of the tent. Now to find the cost of canvas required for the tent we will multiply this by 36.
Cost of canvas required \[ = 14850 \times 36 = Rs.534600\]
This is the final amount.

So total cost of tent is \[Rs.534600\]

Note:
Please note that we calculated the curved surface area because the base and top of the cylinder is not to be built. Also the top of the cylinder is the same as the base of the cone. So for cones also we found the curved surface area. Also the radius of the cylinder is the radius of the cone also. So in calculations we have taken the radius of both the shapes the same. Also don’t forget to write the units.