Question

arc length and area of a sector
the free - throw line on an ncaa basketball court is 12 ft
wide. in international competition, it is only about
11.81 ft. how much longer is the half circle above the
free - throw line on the ncaa court?
in a circle of radius 2 miles, the length of the arc that subtends a central angle of 1
radians is miles.
question help: worked example 1
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Explanation:

Step1: Use arc - length formula

The formula for the arc - length \(s = r\theta\). For a half - circle, \(\theta=\pi\) radians.

Step2: Calculate arc - lengths for both cases

For the NCAA court: The diameter \(d = 12\) ft, so the radius \(r=\frac{d}{2}=6\) ft. Using \(s = r\theta\), \(s_{NCAA}=6\pi\) ft.
For the international competition: The diameter \(d = 11.81\) ft, so the radius \(r=\frac{d}{2}=5.905\) ft. Using \(s = r\theta\), \(s_{int}=5.905\pi\) ft.

Step3: Find the difference

\(\Delta s=s_{NCAA}-s_{int}=(6\pi - 5.905\pi)\) ft. Factor out \(\pi\): \(\Delta s=(6 - 5.905)\pi\) ft. Since \(6-5.905 = 0.095\), and \(\pi\approx3.14\), then \(\Delta s=0.095\times3.14 = 0.2983\) ft.

Answer:

\(0.2983\) ft