Question

A sprinkler system is set up to water the sector shown in the accompanying diagram, with angle ABC measuring 1 radian and radius AB = 20 feet. What is the length of arc AC, in feet?

Answer 1

Answer:

The length of the arc AC is;

$AC=20ft$

Explanation:

Given the radius AB as 20ft.

$AB=20\text{ ft}$

And angle ABC measures 1 radian.

Using the Method 1;

Length of Arc AC is;

$AC=\omega r$

Substituting the angle in radian and the radius;

$\begin{gathered} AC=1\times20\text{ ft} \\ AC=20\text{ ft} \end{gathered}$

The length of Arc AC is 20 ft

Method 2;

We will firstly convert the angle from radian to degree.

As shown below;

$\begin{gathered} \theta=1\text{ rad}\times\frac{180}{\pi} \\ \theta=57.3^0 \end{gathered}$

Then we will apply the formula for calculating the length of an arc;

$C=\frac{\theta}{360}\times2\pi r$

Substituting the value of the angle and the radius.

$\begin{gathered} C=\frac{57.3}{360}\times2\pi\times20ft \\ C=\frac{57.3}{360}\times40\pi ft \\ C=20ft \end{gathered}$

Therefore, the length of the arc AC is;

$AC=20ft$