Question

the figure shows one end of a game court with the three - point line colored red. what is the total length of the three - point line to the nearest foot? the total length of the three - point line is ft. (round to the nearest whole number as needed.)

Explanation:

Step1: Identify the arc - length formula

The formula for the length of an arc of a circle is $s = r\theta$, where $s$ is the arc - length, $r$ is the radius of the circle, and $\theta$ is the central angle in radians. First, convert the angle from degrees to radians. The conversion formula is $\theta_{rad}=\theta_{deg}\times\frac{\pi}{180}$. Given $\theta_{deg}=139.89^{\circ}$, then $\theta_{rad}=139.89\times\frac{\pi}{180}$.

Step2: Calculate the angle in radians

$\theta_{rad}=139.89\times\frac{\pi}{180}\approx139.89\times\frac{3.14159}{180}\approx2.44$ radians. The radius $r = 24.25$ ft.

Step3: Calculate the arc - length

Using the arc - length formula $s = r\theta$, substitute $r = 24.25$ ft and $\theta\approx2.44$ radians. So $s=24.25\times2.44 = 59.17$ ft.

Step4: Add the two straight - line segments

The two straight - line segments each have a length of 14 ft. The total length of the two straight - line segments is $2\times14=28$ ft.

Step5: Calculate the total length of the three - point line

The total length $L$ of the three - point line is the sum of the arc - length and the length of the two straight - line segments. $L=s + 28=59.17+28=87.17\approx87$ ft.

Answer:

87