Chapter 1.2, Problem 1E

(a)

To determine

To find:The formula for the area $A$ of a circle as a function of its diameter $d$ .

Answer to Problem 1E

The formula for the area $A$ of a circle as a function of its diameter $d$ is $A=\frac{\pi d^2}{4}$ .

Explanation of Solution

Given information: The area of the circle is $A$ and diameter is $d$ .

Formulaused:The area of the circle is $A=\pi r^2$ .

Calculation:

As area $A$ of the circle in terms of radius of circle $r$ is $\pi r^2$ . It is known that the radius of the circle is half of its diameter. So,

  $r=\frac{d}{2}$

Substitute $\frac{d}{2}$ for $r$ in the formula for area.

  $\begin{array}{c}A=\pi r^2\\ =\pi {\left(\frac{d}{2}\right)}^2\\ =\frac{\pi d^2}{4}\end{array}$

Therefore, the formula for the area $A$ of a circle as a function of its diameter $d$ is $A=\frac{\pi d^2}{4}$ .

(b)

To determine

To find:The area of the circle withdiameter $4\text{in}$ .

Answer to Problem 1E

The area of the circle with diameter $4\text{in}$ is $4\pi {\text{in}}^2$ .

Explanation of Solution

Given information:The diameter of the circle is $4\text{in}$ .

Calculation:

As calculated in part(a), the formula for the area $A$ of a circle as a function of its diameter $d$ is:

  $A=\frac{\pi d^2}{4}$

Substitute $4$ for $d$ in the above formula.

  $\begin{array}{c}\text{A}=\frac{\pi {\text{d}}^2}{4}\\ =\frac{\pi {\left(4\text{in}\right)}^2}{4}\\ =4\pi {\text{in}}^2\end{array}$

Therefore, the area of the circle with diameter $4\text{in}$ is $4\pi {\text{in}}^2$ .