Question

What length of 5 m wide cloth will be required to make a conical tent, the radius of whose base is 7 m and whose height is 24 m?

Answer 1

Hint: We will first find the curved surface area of the conical tent. After that, we will just divide it by the width of the given cloth to find its length required.

Complete step-by-step answer:
We are given that we need to make a tent using the cloth. This means that we need to find the curved surface area of the tent because that is where the cloth is going to be.
Since, we know that the curved surface area of a cone is given by the following expression:-
$ \Rightarrow CSA = \pi rl$, where r is the radius of the base and l is the slant height of the cone.
Here, we know that l is given by the following expression:-
$ \Rightarrow l = \sqrt {{r^2} + {h^2}} $, where r is the radius of the base and h is the height of the cone.
Now, let us first find the slant height of the given cone to us.
Since, we are given the radius as 7 m and height as 24 m.
Therefore, we have a slant height of $l = \sqrt {{7^2} + {{24}^2}} m$.
Simplifying the calculations inside the square root:-
$ \Rightarrow l = \left( {\sqrt {49 + 576} } \right)m$
Simplifying the calculations further inside the square root:-
$ \Rightarrow l = \left( {\sqrt {625} } \right)m$
So, the slant height is 25 m.
Now, let us find the curved surface area.
$ \Rightarrow CSA = \dfrac{{22}}{7} \times 7 \times 25$ square meter
Doing the mentioned calculations in the above expression, we get:-
$ \Rightarrow $Curved surface area = 550 sq meter.
Now, the area of cloth we use must be equal to the just found curved surface area.
Therefore, $l \times 5 = 550$ (Because area of a rectangle is given by the product of its length and breadth)
Dividing the whole equation by 5, we get:-
$ \Rightarrow l = \dfrac{{550}}{5}$
On simplification, we get:- Length = 110 m.

Hence, the length of the cloth required is 110 m.

Note:
The students must note that we used the curved surface area of a cone, because the cloth in a tent is there only unless it is mentioned that the tent has a base, we use the curved surface area only. If the base as well, find its area and add it to the curved surface area and then proceed further.
Do not forget to put units after length and area, because just writing numbers would not make any sense unless there are units written there over with them.