Cowboy Clint wants to build a dirt road from his ranch to the highway so that he can drive to the city in the shortest amount of time.

Question

Cowboy Clint wants to build a dirt road from his ranch to the highway so that he can drive to the city in the shortest amount of time. The perpendicular distance from the ranch to the highway is 4 miles, and the city is located 9 miles down the highway. Where should Clint join the dirt road to the highway if the speed limit is 20 mph on the dirt road and 55 mph on the highway?

Answer 1

Draw a figure with the farm on the y axis and the city on the x axis.

Let x be the point where the dirt road joins the highway.

Then (1/20)sqrt(16+x²)+(1/55)(9-x) = T, i.e. time = distance/rate

dT/dx = (1/20)(x/sqrt(16+x²) - (1/55)

At the critical point

x²/(16+x²) = (20/55)² = c²

x²(1-c²) = 16c²

x= 4c/sqrt(1-c²)

Answer 2

Joining the highway:

desmos.com/calculator/a3ybjiwpwe

Table of values:

desmos.com/calculator/wmmmshj3zl