Question

How Many meters of cloth 2.5m wide will be required to make a conical tent whose radius is 7m and height is 24m?

Answer 1

Hint: The term conical tent means that they only need a surface in cone shape no base is required. So, first find the surface area of shape of the cone excluding the base. Now use Pythagora's theorem to find the slant height of cone. Pythagora's theorem must be applied on the lengths of radius, height correspondingly. After finding the slant height substitute this into the formula. Now you get the area of cloth you need. So, the area divided by width gives you the length of cloth. Which is the required result. Total surface area of cone is given by:
$A=\pi rl+\pi {{r}^{2}}$

Complete step-by-step answer:
Slant Height: The length of line from open to the point on a circle is called slant height. It is denoted by l.
By looking at the cone we can say that the surface area of the cone is the surface area of the conical surface area of the box.
As base is circle, we get it as $\pi {{r}^{2}}$

So, surface area of conical part is given by:
Let it be A.
A= surface area- $\pi {{r}^{2}}$
We know surface area of a cone is given by:
$S=\pi rl+\pi {{r}^{2}}$
By substituting this we get the value of A as:
$A=\pi rl$
By looking at the diagram of cone carefully we can get a triangle formed by h,r,l as:

By applying Pythagoras theorem to this, we get:
${{l}^{2}}={{h}^{2}}+{{r}^{2}}$
By applying square root on both side of equation, we get it as:
$\sqrt{{{l}^{2}}}=\sqrt{{{h}^{2}}+{{r}^{2}}}$
By substituting values of h, r we get the value of l as:
$l=\sqrt{{{\left( 24 \right)}^{2}}+{{7}^{2}}}=\sqrt{576+49}=\sqrt{625}$
By simplifying this equation, we get value of l as:
l=25 m
By substituting this into formula of A, we get it as:
$A=\pi \times 7\times 25=155\pi $
By equating A= length × width and substituting width, $\pi =\dfrac{22}{7}$, we get:
$length\times 2.5=155\times \dfrac{22}{7}=550$
By simplifying the above equation, we get value of length as:
Length= 220m.
So, we need a cloth of 220m which is 2.5m wide to get a conical tent of height 24m and radius 7m.

Note: Substitute r, n in correct places. Generally, students confuse between r, n. while applying Pythagoras theorem be careful that you substitute the correct term as hypotenuses and correct term as adjacent sides. The formula $\pi rl+\pi {{r}^{2}}$ is predefined if you want proof you must use the concept of integration to prove that the surface area formula is this.