Chapter 12.8, Problem 25PPS

To determine

To determine the volume of the larger tent.

Answer to Problem 25PPS

  $89.8{\text{ feet}}^3$

Explanation of Solution

Given:

Two tents are in the shape of hemispheres, with circular floors.

The ratio of their floor areas is 9:12.25.

The diameter of the smaller tent is 6 feet.

Formula used:

 Volume of a hemisphere $=\frac{2}{3}\pi r^3$

Calculation:

Two tents are in the shape of hemispheres, with circular floors.

The ratio of their floor areas is 9:12.25.

The diameter of the smaller tent is 6 feet.

The scale factor of surface area of both the tent is,

  $\Rightarrow \frac{\text{Surface area of smaller tent}}{\text{Surface area of larger tent}}=\frac{9}{12.25}$

  $\Rightarrow \frac{\text{Surface area of smaller tent}}{\text{Surface area of larger tent}}=\frac{3^2}{{3.5}^2}$

Scale factor of surface area of both the tent is 3:3.5.

Now, the radius of larger tent,

  $\Rightarrow \frac{\text{Radius of smaller tent}}{\text{Radius of larger tent}}=\frac{3}{3.5}$

As, scale factor is same for all.

  $\Rightarrow \frac{3}{\text{Radius of larger tent}}=\frac{3}{3.5}$

  $\Rightarrow \text{Radius of larger tent}=\frac{3.5\times 3}{3}$

  $\Rightarrow \text{Radius of larger tent}=3.5$ feet

Now, find the volume of hemisphere. 

Volume of a hemisphere $=\frac{2}{3}\pi r^3$

  $=\frac{2}{3}\pi r^3$

  $=\frac{2}{3}\times \pi \times {(3.5)}^3$

  $=\frac{2}{3}\times 134.695$

  $=89.797{\text{ feet}}^3$

  $\approx 89.8{\text{ feet}}^3$

Conclusion:

Therefore, the volume of the larger tent is $\approx 89.8{\text{ feet}}^3$ .