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A. $ 20\;{\text{m}} $

B. $ 15\;{\text{m}} $

C. $ 12\;{\text{m}} $

D. $ 30\;{\text{m}} $

Answer

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Given:

The total number of people in the ceremony is $ 150 $ .

The space required on the ground for each person is $ 4\;{\text{sq}}{\text{.}}\;{\text{meters}} $ .

Volume of air required by one person is $ 20\;{\text{cubic}}\;{\text{meters}} $ .

The following is the schematic diagram of the conical tent.

For the conical tent, the area of the base of the tent will be,

$

\Rightarrow A = 150 \times 4\;{\text{sq}}{\text{.}}\;{\text{meters}}\\

\Rightarrow A = 600\;{\text{sq}}{\text{.}}\,\;{\text{meters}}

$

The formula for the area of the base of the cone is $ \pi {r^2} $ .

Therefore, \[\pi {r^2} = 600\;{\text{sq}}{\text{.}}\;{\text{meters}}\].

So, the above expression of area can be written as:

$

{r^2} = \dfrac{{600\;{\text{sq}}{\text{.}}\;{\text{meters}}}}{\pi }\\

{r^2} = \dfrac{{2100}}{{11}}\;{\text{sq}}{\text{.}}\;{\text{meters}}

$

Volume of air required by 150 people will be,

$ v = 150 \times 20\;{\text{cubic meters}} = 3000\;{\text{cubic meters}} $

The formula for the volume of cone is $ v = \dfrac{1}{3}\pi {r^2}h $ .

Substitute $ 3000\;{\text{cubicmeters}} $ for $ v $ and $ \dfrac{{2100}}{{11}}\;{\text{sq}}{\text{.}}\;{\text{meters}} $ for $ {r^2} $ in the above expressions.

$ 3000\;{\text{cubic meters}} = \dfrac{1}{3}\pi \times \dfrac{{2100}}{{11}}\;{\text{sq}}{\text{.}}\;{\text{meters}} \times h $

Rearrange the above expression.

$

\Rightarrow h=\dfrac{{3000\;{\text{cubic meters}}}}{{\dfrac{1}{3}\pi \times \dfrac{{2100}}{{11}}\;{\text{sq}}{\text{.}}\;{\text{meters}}}}\\

h=\dfrac{{3000}}{{\dfrac{1}{3} \times \dfrac{{22}}{7} \times \dfrac{{2100}}{{11}}}}{\text{meters}}\\

h = 15\;{\text{meters}}

$

Hence, the height of the conical tent will be $ 15\;{\text{m}} $ .