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A cloth having an area of $165\; m^{2}$ is shaped into the form of a conical tent of radius $5\; m$

How many students can sit in the tent if a student, on an average, occupies $\frac{5}{7}m^{2}$ on the ground?

Find the volume of the cone.

Answers (1)

Answer : (i) 110 students

(ii) $241.73\; m^{3}$

We have, Area of cloth $=165\; m^{2}$

This cloth is shaped in the form of a conical tent.

Radius of conical tent = 5 cm

So, we have:

Area of cloth = Curved surface area of cone

Curved surface area of cone is given as $\pi rl$ Where, r = radius of a cone

l = slant height of a cone

So curved surface area of conical tent,

$165=\frac{22}{7}\times 5 \times l$

$\Rightarrow l=\frac{165 \times 7}{22 \times 5}=\frac{21}{2}=10.5\; m$

(i) Now it is given that area covered by 1 student $=\frac{5}{7}m^{2}$

$\text {So the number of students}=\frac{\text {Area of a circular base of a cone}}{\text {Area covered by 1 student}}$

$\Rightarrow \text {Number of student}=\frac{\pi r^{2}}{\frac{5}{7}}=\frac{\left ( \frac{22}{7}\times 5^{2} \right )}{\frac{5}{7}}=110$

So 110 students can sit in the tent.

(ii) Volume of cone : $\frac{1}{3}\pi r^{2}h$

For a right circular cone, we have

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

$\Rightarrow 5^{2}+h^{2}=\left ( 10.5 \right )^{2}$

$\Rightarrow 25+h^{2}=110.25$

$\Rightarrow h^{2}=110.25-25=85.25$

$h=\sqrt{85.25}=9.23\; m$

So, volume $=\frac{1}{3}\times \frac{22}{7}\times 5^{2}\times 9.23$

$=\frac{5076.5}{21}=241.73\; m^{3}$

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