Question

find the distance along an arc on the surface of earth that subtends a central angle of 24 minutes (1 minute = \\( \frac { 1 } { 60 } \\) degree). the radius of earth is 3,960 miles. round to the nearest hundredth of a mile. miles

Explanation:

Step1: Convert minutes to degrees

Since \(1\) minute \(=\frac{1}{60}\) degree, for \(24\) minutes, the angle in degrees is \(\theta = 24\times\frac{1}{60}=0.4\) degrees.

Step2: Convert degrees to radians

We use the conversion formula \(\text{radians}=\frac{\pi}{180}\times\text{degrees}\). So \(\theta = 0.4\times\frac{\pi}{180}=\frac{\pi}{450}\) radians.

Step3: Use the arc - length formula

The arc - length formula is \(s = r\theta\), where \(r = 3960\) miles (radius of the Earth) and \(\theta=\frac{\pi}{450}\) radians. Then \(s=3960\times\frac{\pi}{450}\).

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Answer:

\(27.65\)