A bicyclist is at point A on a paved road and must ride to p

A bicyclist is at point A on a paved road and must ride to point C on another paved road. The two roads meet at an angle of 31 degree at point B. The distance from A to B is 25 mi, and the distance from B to C is 12 mi (see the figure). If the bicyclist can ride 16 mph on the paved roads and 12.6 mph off road, would it be faster for the bicyclist to ride from A to C on the paved roads or to ride a direct line from A to C off road? The time required to ride off road is approximately hr min, whereas the time required to ride on the paved road is hr min. Therefore, the rider should.

Solution

Apply cosine rule to find length of road direct A to C :

AC^2 = AB^2 + BC^2 - 2AB*BCcos31

= 25^2 + 12^2 -2*25*12cos31

AC = 15.95 miles

Time taken to ride off road = distance/speed = 15.95/12.6 = 1.27 = 1hr and 16 minutes

Time taken to ride paved road = distance/ speed = (25 +12)/16 = 2.31 = 2hrs + 19 minur

So, the rider should ride off road