Chapter 1.1, Problem 53E

To determine

To find the area of the base of the given solid.

Answer to Problem 53E

  $12\pi i{n}^2$

Explanation of Solution

Given:

  

Formula used:

Volume of a cylinder:

  $V=\pi r^{2}h$

Area of circle:

  $A=\pi r^2$

Calculation:

The given image is a cylinder. The base of the cylinder is a circle.

Therefore, the area of the base of the cylinder is the area of the circle.

From the given image,

Volume $=84\pi i{n}^3$

Height of the cylinder $=7in$

Substitute the known values in formula (volume of the cylinder).

  $\begin{array}{l}\Rightarrow 84\pi =\pi r^{2}7\\ \Rightarrow 84=r^{2}7[\text{Dividing by }\pi \text{ on both sides}]\\ \Rightarrow 12=r^2\text{ [Dividing by 7 on both sides}]\\ \Rightarrow \sqrt{12}=r\\ \Rightarrow \sqrt{4\times 3}=r\\ \Rightarrow r=2\sqrt{3}\end{array}$

Substitute the value of r in the area of circle to find the area of the base.

  $\begin{array}{l}$ \Rightarrow A=\pi {(2\sqrt{3})}^2$\Rightarrow A=\pi (4\times 3)\\ \Rightarrow A=12\pi \end{array}$

Conclusion:

Therefore, the area of the base is $12\pi i{n}^2.$