Question

A conical tent with base-radius 7m and height 24 m is made from 5m wide canvas. The length of the canvas used is$(\pi = \dfrac{{22}}{7})$.
$
A. {\text{ 100m}} \\
B. {\text{ 105m}} \\
C. {\text{ 110m}} \\
D. {\text{ 115m}} \\
 $

Answer 1

Hint: In this question first we need to calculate the area of canvas in terms of unknown length. Then calculate the curved surface area of the cone and equate them to get the length of canvas used.

Complete Step-by-step answer:
Since, Canvas is rectangular in shape and let its dimensions be length and wide $w = 5{\text{m}}$.
We know,
Area of rectangle =length$ \times $wide

Then, area of canvas=${\text{length}} \times 5{{\text{m}}^2}$ eq.1
Now, the tent is in the shape of a cone. Let its dimensions be radius (r=7m) and height (h=24m) and its slant height be l.

We know the curved surface area of cone = $\pi rl$
Now, for a cone relation between the radius, slant height and height is given by
$ \Rightarrow l = \sqrt {{h^2} + {r^2}} $
Using above relation, we can find the value of slant height of tent
$
   \Rightarrow l = \sqrt {{{(24)}^2} + {{(7)}^2}} \\
   \Rightarrow l = \sqrt {625} \\
   \Rightarrow l = 25{\text{m}} \\
 $
Then,
The curved surface area of tent = curved surface area of come
                                                        =$\pi rl$
                                                      = $\dfrac{{22}}{7} \times 7 \times 5{\text{ \{ take }}\pi {\text{ = }}\dfrac{{22}}{7}\} $
                                                    =$550{{\text{m}}^2}{\text{ eq}}{\text{.2}}$
It is given that conical tents are made from canvas.
Then,
${\text{Area of canvas = Curved surface area of tent}}$ eq.3
Then from eq. 1 and eq.2 we can write eq.3 as
$
   \Rightarrow {\text{length}} \times {\text{5 = 550}} \\
   \Rightarrow {\text{ length = 110m }} \\
 $
Therefore, the length of the canvas used is 110 m.
Hence, option C is correct.

Note: Whenever you get this type of question the key concept to solve this is to learn the formulas of different shapes like in this case we require the formula of area of rectangle(length$ \times $wide) and the curved surface area of cone ($\pi rl$).