Question

The diameter of a circle is 8 miles. What is the angle measure of an arc 3​ miles long?

Answer 1

• Radius=8/2=4mi
• length of arc=l=3mi

Now

`\\ \rm\rightarrowtail \theta=(l)/(r)`

`\\ \rm\rightarrowtail l=r\theta`

`\\ \rm\rightarrowtail 3=4\theta`

`\\ \rm\rightarrowtail \theta=(3)/(4)^c`

Answer 2

`\sf \theta=\frac34 \ radians=42.97 \textdegree \ (nearest \ hundredth)`

Explanation:

Formulae

`\sf radius (r)=\frac12 d`

(where d is the diameter of a circle)

`\sf arc \ length =r\theta`

(where r is the radius and
`\theta` is measured in radians)

Calculation

Given:

• 
  `\sf radius (r)=\frac12 \cdot 8=4 \ mi`
• arc length = 3 mi

Substituting these values into the formula for arc length:

`\implies \sf 3 =4\theta`

`\implies \sf \theta=\frac34 \ radians`

To convert radians to degrees use

`\sf 1 \ rad \cdot (180\texrdegree)/(\pi)`

`\implies \sf \frac34 \cdot (180\texrdegree)/(\pi)=42.97183463...\textdegree`