Question

A tent is in the form of a right circular cone \[10.5\] m high, the diameter of the base being \[13\] m. If \[8\] men are in the tent, find the average number of cubic meters of air space per man.

Answer 1

Hint: To find the average number of cubic meters of air space per man, at first, we need to find the total air space of the tent. It means the volume of the tent. Then, by dividing the total air space by the number of men we will get the average number of cubic meters of air space per man.
We know that, if $h$ & $r$ be the height and the radius of the cone respectively, then, the volume of the cone is \[V = \dfrac{1}{3}\pi {r^2}h\]

Complete step-by-step answer:
It is given that; height of the right circular cone is \[10.5\] m. The diameter of the base is \[13\] m. There are \[8\] men in the tent.
We have to find the average number of cubic meters of air space per man.
First, we have to find the volume of the cone.
Since, the diameter of the base is \[13\] m.
So, the radius of the base is \[\dfrac{{13}}{2}\] m\[ = 6.5\] m.
Let us consider, $h$ & $r$ be the height and the radius of the cone respectively. Then, the volume of the cone is \[V = \dfrac{1}{3}\pi {r^2}h\]
Substitute the values of height and the radius in the above formula of the volume we get,
\[V = \dfrac{1}{3} \times \dfrac{{22}}{7} \times {6.5^2} \times 10.5\] ${m^3}$
Let us now simplify the volume we get,
The volume is \[V = 464.75\] ${m^3}$.
Now, \[8\] men occupied \[464.75\] ${m^3}$ of air space.
To find the air space occupied by one man we should divide the volume by 8.
So, \[1\] man occupied \[\dfrac{{464.75}}{8}\] ${m^3}$ of air space.
By simplifying the division we get,
\[1\] man occupied \[58.09375\] ${m^3}$ of air space.
Hence, each man in average occupied \[58.09375\] ${m^3}$ of air space in the tent.

Note: Average value is the typical central value of the set of a data. To find the average we will sum up the value of the set and then divide it by the numbers of data. Since it is given that the total space is occupied by one man we should divide the total space occupied by 8 men by the total number of men.