In the setting of Examples 2 and 3, replace 30 and 50 with general distances D and L, respectively. Also, let r denote the speed along the road, and h denote the speed along the highway. Show that the travel–time function T(x) has a critical point at x = √D (h/r)² − 1.

Example 2

Example 3

EXAMPLE 2 Minimizing Travel Time Your task is to build a road joining the small town of Calverton to Route 1 to enable drivers to reach Capital City in the shortest time (Figure 3). How should this be done if the speed limit is 60 km/hour on the road and 110 km/h on Route 1? The perpendicular distance from Calverton to Route 1 is 30 km, and Capital City is 50 km down Route 1.