Question

circle k has a diameter of 6 feet. what is the length of (overarc{jl})? write an exact, simplified answer. feet

Explanation:

Step1: Find the radius of the circle

The diameter of circle \( K \) is 6 feet, so the radius \( r \) is half of the diameter.
\( r=\frac{6}{2}=3 \) feet.

Step2: Recall the formula for the length of an arc

The formula for the length of an arc \( s \) is \( s = \frac{\theta}{360^\circ}\times 2\pi r \), where \( \theta \) is the central angle in degrees and \( r \) is the radius of the circle. Here, \( \theta = 45^\circ \) and \( r = 3 \) feet.

Step3: Substitute the values into the formula

Substitute \( \theta = 45^\circ \) and \( r = 3 \) into the arc length formula:
\( s=\frac{45^\circ}{360^\circ}\times 2\pi\times 3 \)

Step4: Simplify the expression

First, simplify \( \frac{45}{360}=\frac{1}{8} \). Then, \( 2\pi\times 3 = 6\pi \). So, \( s=\frac{1}{8}\times 6\pi=\frac{6\pi}{8}=\frac{3\pi}{4} \).

Answer:

\( \frac{3\pi}{4} \)