Question

the radius of a circle is 2 miles. what is the length of a 126° arc? give the exact answer in simplest form. miles

Explanation:

Step1: Recall arc length formula

The formula for the length of an arc of a circle is \( s=\frac{\theta}{360^\circ}\times2\pi r \), where \( \theta \) is the central angle in degrees and \( r \) is the radius of the circle.

Step2: Identify given values

We are given that \( r = 2 \) miles and \( \theta=126^\circ \).

Step3: Substitute values into the formula

Substitute \( r = 2 \) and \( \theta = 126^\circ \) into the formula:
\( s=\frac{126^\circ}{360^\circ}\times2\pi\times2 \)
First, simplify \( \frac{126}{360}=\frac{7}{20} \) (dividing numerator and denominator by 18). Then, \( 2\times2 = 4 \), so we have \( s=\frac{7}{20}\times4\pi \).
Simplify \( \frac{7}{20}\times4=\frac{7}{5} \) (since \( 4\div20=\frac{1}{5} \), then \( 7\times\frac{1}{5}=\frac{7}{5} \)? Wait, no: \( \frac{7}{20}\times4=\frac{7\times4}{20}=\frac{28}{20}=\frac{7}{5} \)? Wait, no, 4 and 20 can be simplified: 4/20 = 1/5, so 7(1/5)=7/5? Wait, no, 2πr is the circumference, which is 2π2 = 4π. Then the arc length is (126/360)4π. 126/360 = 7/20, so (7/20)4π = (7*4π)/20 = (28π)/20 = (7π)/5. Wait, let's do it again:

\( s=\frac{126}{360}\times2\pi\times2 \)

\( 2\pi\times2 = 4\pi \), so \( s=\frac{126}{360}\times4\pi \)

Simplify \( \frac{126}{360} \): divide numerator and denominator by 18: 126÷18=7, 360÷18=20. So \( \frac{7}{20} \). Then \( \frac{7}{20}\times4\pi=\frac{7\times4\pi}{20}=\frac{28\pi}{20}=\frac{7\pi}{5} \) (dividing numerator and denominator by 4: 28÷4=7, 20÷4=5).

Answer:

\( \frac{7}{5}\pi \) (or \( \frac{7\pi}{5} \))