For us to be able to determine the distance along an arc on the surface of the earth, we will be using the following formula:
[tex]\text{ S = r}\theta[/tex]
Where,
S = arc length
r = radius (radius of the earth)
θ = central angle (in radian)
Given:
r = 3960 miles
θ = 48 mins.
a.) Let's convert the given measure of the central angle to radian.
[tex]\theta=48mins.\text{ = (48 mins.) x }\frac{1^{\circ}}{(60\text{ mins.})}\text{ = }\frac{48}{60}(1^{\circ})[/tex][tex]\theta\text{ = }\frac{4}{5}^{\circ}[/tex][tex]\text{ }\theta_{radian}\text{ = }\theta_{degrees}\text{ x }\frac{\pi}{180^{\circ}}[/tex][tex]\text{ }\theta_{radian}\text{ = }\frac{4}{5}\text{ x }\frac{\pi}{180}\text{ = }\frac{4\pi}{900}\text{ = }\frac{\pi}{225}\text{ radians}[/tex]
b.) Let's now determine the distance (arc length).
[tex]\text{ S = r}\theta[/tex][tex]\text{ S = (3960)(}\frac{\pi}{225}\text{ ) = }\frac{3960\pi}{225}\text{ miles = 17.6}\pi\text{ miles = 55.2920307 }\approx\text{ 55.292 miles}[/tex]
Therefore, the answer is 55.292 miles.
