Question

for a circle with a radius of 6 meters, what is the measurement of a central angle (in degrees) subtended by an arc with a length of 5/2 pi meters

Answer 1

`\bf \textit{arc's length}\\\\ s=\cfrac{\pi r\theta }{180}\quad \begin{cases} r=radius\\ \theta =angle~in\\ \qquad degrees\\ ------\\ r=6\\ s=(5\pi )/(2) \end{cases}\implies \cfrac{5\pi }{2}=\cfrac{\pi \cdot 6\cdot \theta }{180}\implies \cfrac{5\pi }{2}=\cfrac{\pi \theta }{30} \\\\\\ \cfrac{5\underline{\pi }\cdot 30}{2\underline{\pi} }=\theta \implies 75=\theta`

Answer 2

`75\°`

Explanation:

we know that

The circumference of a circle is equal to

`C=2\pi r`

In this problem we have

`r=6\ m`

substitute the value

`C=2\pi (6)=12\pi\ m`

Remember that

`360\°` subtends the complete circle of length arc
`12\pi\ m`

so

by proportion

Find the central angle for an arc length of
`(5\pi)/(2)\ m`

`(360)/(12\pi)(degrees)/(m)=(x)/((5\pi)/(2))(degrees)/(m) \\ \\ x=(5/2)*360/12\\ \\x=75\°`