Question

Monica has a piece of canvas whose area is 551 m 2 .She uses itto have a conical tent made, with base radius of 7 m. Assuming that all the stitching margins and the wastage incurred while cutting, amounts to approximately 1 m 2 , find the volume of the tent that can be made with it.

Answer 1

Given :

Monica has a piece of canvas whose area is 551 m 2 .She uses itto have a conical tent made, with base radius of 7 m. Assuming that all the stitching margins and the wastage incurred while cutting, amounts to approximately 1 m 2

To prove :

Find the volume of the tent .

solution :

[ Take π = 22/7 ]

radius (r) = 7 cm

Curved surface area of the conical tent = Area of the of the canvas - area of the wastage used

= 551 m² - 1 m²

= 550 m²

=> πrl = 550 m²

$\boxed { Curved \: surface \: Area \: = πrl }$

$\implies \frac{22}{7}\times 7\times l = 550$

=> l = ( 550 )/22

=> l = 25 m

$\boxed { h^{2} = l^{2}-r^{2} }$

=> h² = 25² - 7²

=> h² = 625 - 49

=> h² = 576

=> h = √576

=> h = √24²

=> h = 24 m

Now ,

$\boxed { Volume \: of \: a : cone = \frac{1}{3}\times π\times r^{2}\times h}$

V = (1/3) × (22/7) × 7² × 24

= (22×7×7×24)/(3×7)

= 22×7×8

= 1232 m³

Therefore,

Volume of the conical tent (V) = 1232 m³

••••

Answer 2

Answer:-

We have ,

Area of Canvas = 551 m²

Area of Canvas lost in wastage = 1 m²

$\therefore$ Area of Canvas used in making tent = ( 551 - 1 ) m² = 550 m²

$\implies$ Surface area of the cone = 550 m²

We have,

r = radius of the base of the cone = 7 m

$\therefore$ $\mathfrak\red{Surface\:\:area=\:\:550\:m^2}$

$\implies$ $\sf{\pi\:rl=550}$

$\implies$ $\sf\dfrac{22}{7}\times\:7\times\:l=500$

$\implies$ $\sf\blue{l=25m}$

_________________________

Let h be the height of the cone. Then,

$\sf{l^2=r^2+h^2}$

$\implies$ $\sf{h=}$$\sf\sqrt{l^2-r^2}$

$\leadsto$ $\sf\sqrt{25^2-7^2}$

$\leadsto$ $\sf\pink{24m}$

________________________

$\therefore$ Volume of the cone

$\sf \: = \frac{1}{3} \pi {r}^{2} h \\ \\ \sf \: = \frac{1}{3} \times \frac{22}{7} \times 7 \times 7 \times 24 \: {m}^{3} \\ \\ \sf \: =1232 \: {m}^{3}$