Question

course challenge google classroom microsoft teams a circle has a radius of 4. an arc in this circle has a central angle of 288°. what is the length of the arc? either enter an exact answer in terms of π or use 3.14 for π and enter your answer as a decimal.

Explanation:

Step1: Recall arc length formula

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

Step2: Substitute given values

We know that \( \theta = 288^\circ \) and \( r = 4 \). Substitute these values into the formula:
\( L=\frac{288^\circ}{360^\circ}\times2\pi\times4 \)
First, simplify \( \frac{288}{360}=\frac{4}{5} \) (dividing numerator and denominator by 72). Then:
\( L=\frac{4}{5}\times8\pi \) (since \( 2\times4 = 8 \))
\( L=\frac{32}{5}\pi \) or if we use \( \pi = 3.14 \), we have \( L=\frac{32}{5}\times3.14 \)

Step3: Calculate the decimal value

\( \frac{32}{5}=6.4 \), so \( 6.4\times3.14 = 20.096 \)
Or in terms of \( \pi \), \( \frac{32}{5}\pi=\frac{32\pi}{5} \) (or \( 6.4\pi \))

Answer:

If in terms of \( \pi \): \( \frac{32\pi}{5} \) (or \( 6.4\pi \)); if as a decimal (using \( \pi = 3.14 \)): \( 20.096 \)