To determine the length of a wire that forms an arc of a circle with a radius of [tex]\(10.5\)[/tex] meters and a central angle of [tex]\(150^\circ\)[/tex], follow these steps:
1. Convert the angle from degrees to radians:
The formula to convert degrees to radians is:
[tex]\[
\text{radians} = \text{degrees} \times \left(\frac{\pi}{180}\right)
\][/tex]
Given:
[tex]\[
\text{angle in degrees} = 150^\circ
\][/tex]
Use [tex]\(\pi = \frac{22}{7}\)[/tex]:
[tex]\[
\text{angle in radians} = 150 \times \left(\frac{22}{7 \times 180}\right)
\][/tex]
2. Angle conversion:
Simplify the angle conversion:
[tex]\[
\text{angle in radians} = 150 \times \left(\frac{22}{1260}\right)
= 150 \times \frac{11}{630}
= 150 \times \frac{1}{60}
= 2.5 \text{ radians}
\][/tex]
3. Calculate the length of the arc:
The length [tex]\(S\)[/tex] of an arc is given by the formula:
[tex]\[
S = r \theta
\][/tex]
where [tex]\(r\)[/tex] is the radius and [tex]\(\theta\)[/tex] is the angle in radians. Given:
[tex]\[
r = 10.5 \, \text{meters}
\][/tex]
and from the previous step,
[tex]\[
\theta = 2.619047619047619 \, \text{radians}
\][/tex]
4. Compute the arc length:
Substitute the values into the formula:
[tex]\[
S = 10.5 \times 2.619047619047619
\][/tex]
5. Simplify the expression:
Compute:
[tex]\[
S = 27.5 \, \text{meters}
\][/tex]
Hence, the length of the wire is [tex]\(27.5\)[/tex] meters.
