Chapter 6.2, Problem 2YT

YOUR TURN Suppose the animal in Example 2 can run only 40 m per minute through the woods. Find the path that will get it to the spring in the minimum time.

EXAMPLE 2 Minimizing Time

A thirsty animal wants to get to a spring as soon as possible. It can get to the spring by travelling east along a clear path for 300 meters, and then north through the woods for 800 meters. The animal can run 160 meters per minute along the clear path, but only 70 meters per minute through the woods. Running directly through the woods towards the spring minimizes the distance, but the animal will be going slowly the whole time. It could instead run 300 meters along the clear path before entering the woods, maximizing the total distance but minimizing the time in the woods. Perhaps the fastest route is the combination, as shown in Figure 7. Find the path that will get the animal to the spring in the minimum time. (Although it may seem odd for an animal to follow the path given by calculus, a mathematician has shown that a dog chasing a ball does exactly that. Source: The College Math Journal.)