Question

A 5m wide cloth is used to make a conical tent of base diameter $14m$ and height $24m$. Find the cost of cloth used at the rate of $Rs.25$ per meter. (Use $\pi = \dfrac{{22}}{7}$ )

Answer 1

Hint: In the given question we have to find the cost of the cloth and for that, we need to find the length of the cloth. As already given in the question, that cloth is used to make a conical (conical means that it is in the shape of a cone) tent that means the curved surface area of the conical tent is equal to the area of cloth. We can find the slant height of the conical tent using the information provided in the question. And from that, we can calculate the curved surface area of a conical tent which is equal to the area of cloth. From the area of cloth, we can calculate the length of cloth by using the formula of area of rectangle. Now, we have found the length of cloth so we can easily calculate the cost of cloth.
Formulae used:
The curved surface area of cone = $\pi rl$
Slant height of cone $l = \sqrt {{r^2} + {h^2}} $.
Area of rectangle = length $×$ breadth

Complete step-by-step solution:

Given, Diameter of the conical tent = $14m$
$\therefore r = 7m$
Now, the tent is in the shape of a cone. Let its dimensions be radius $\left( {r = 7m} \right)$ and height $\left( {h = 24m} \right)$ and its slant height be $l$.
Now, for a cone relation between the radius, slant height and height is given by
$ \Rightarrow l = \sqrt {{r^2} + {h^2}} $
Using above relation, we can find the value of slant height of tent
$ \Rightarrow l = \sqrt {{r^2} + {h^2}} $
Substitute the values of radius and height.
$ \Rightarrow l = \sqrt {{7^2} + {{24}^2}} m$
On solving square, we get
$ \Rightarrow l = \sqrt {49 + 576} m$
$ \Rightarrow l = \sqrt {625} m$
On solving square root, we get
$ \Rightarrow l = 25m$
Then, the curved surface area of tent = curved surface area of cone = $\pi rl$
$ = \left( {\dfrac{{22}}{7} \times 7 \times 25} \right){m^2}$
$ = 22 \times 25{m^2}$
$ = 550{m^2}$
It is given that conical tent is made from cloth, then
Area of cloth = Curved surface area of tent
Therefore, area of cloth $ = 550{m^2}$
Length ×breadth $ = 550{m^2}$
$ \Rightarrow length \times 5 = 550$
$ \Rightarrow length = \dfrac{{550}}{5}m$
$ \Rightarrow length = 110m$
As already given in the question cost of $1metre$ of cloth = $Rs25$
Cost of $110metre$ of cloth = \[Rs{ 25 \times 110}\]
Therefore, cost of $110metre$ of cloth = \[Rs{\text{ 2750}}\]

Note: The key concept to solve this type of question is to learn the formulas of different shapes. Students must know where to use total surface area and where to use the curved surface area for that to read the question carefully. Write length only in positive value because length can’t be negative. Students should be careful about the unit.