Question

1. use three unit multipliers to convert 40 meters to feet.
2. the area of a circle is 25 in.².

(a) find the radius of the circle.
(b) find the circumference of the circle.

Explanation:

Step1: Recall the formula for the area of a circle

The formula for the area of a circle is \( A=\pi r^{2} \), where \( A \) is the area and \( r \) is the radius. We know that \( A = 25\space\text{in}^2\), so we substitute this value into the formula: \( 25=\pi r^{2} \).

Step2: Solve for the radius \( r \)

First, divide both sides of the equation by \( \pi \): \( r^{2}=\frac{25}{\pi} \). Then, take the square root of both sides. Since the radius is a length (a positive quantity), we consider the positive square root: \( r = \sqrt{\frac{25}{\pi}}=\frac{5}{\sqrt{\pi}}\approx\frac{5}{1.772}\approx2.82\space\text{in} \) (if we use \( \pi\approx3.1416 \), we can also calculate \( r^{2}=\frac{25}{3.1416}\approx7.958 \), then \( r=\sqrt{7.958}\approx2.82\space\text{in} \)).

Step3: Recall the formula for the circumference of a circle

The formula for the circumference of a circle is \( C = 2\pi r \) (or also \( C=\pi d \), where \( d \) is the diameter, and \( d = 2r \)).

Step4: Substitute the radius into the circumference formula

We found that \( r=\frac{5}{\sqrt{\pi}}\space\text{in} \). Substitute this into the circumference formula: \( C = 2\pi\times\frac{5}{\sqrt{\pi}}=2\times5\times\sqrt{\pi}=10\sqrt{\pi}\approx10\times1.772\approx17.72\space\text{in} \) (or if we use \( r\approx2.82\space\text{in} \), then \( C = 2\times3.1416\times2.82\approx17.72\space\text{in} \)).

Part (a)

Step1: Start with the area formula

Given \( A=\pi r^{2}=25 \).

Step2: Solve for \( r \)

\( r^{2}=\frac{25}{\pi} \), so \( r = \sqrt{\frac{25}{\pi}}=\frac{5}{\sqrt{\pi}}\approx2.82\space\text{in} \) (exact form: \( \frac{5}{\sqrt{\pi}}\space\text{in} \) or rationalized as \( \frac{5\sqrt{\pi}}{\pi}\space\text{in} \), approximate decimal value is about \( 2.82\space\text{in} \)).

Part (b)

Step1: Use the circumference formula \( C = 2\pi r \)

We know \( r=\frac{5}{\sqrt{\pi}} \), substitute into the formula: \( C = 2\pi\times\frac{5}{\sqrt{\pi}} \).

Step2: Simplify the expression

Simplify \( 2\pi\times\frac{5}{\sqrt{\pi}}=10\sqrt{\pi}\approx17.72\space\text{in} \) (or using \( r\approx2.82\space\text{in} \), \( C = 2\times3.1416\times2.82\approx17.72\space\text{in} \)).

Answer:

(a)

The radius \( r=\frac{5}{\sqrt{\pi}}\space\text{in}\approx2.82\space\text{in} \) (or \( \frac{5\sqrt{\pi}}{\pi}\space\text{in} \)).

(b)

The circumference \( C = 10\sqrt{\pi}\space\text{in}\approx17.72\space\text{in} \).