EXAMPLE 2 Minimizing Time A math professor participating in the sport of orienteering must get to a specific tree in the woods as fast as possible. He can get there by traveling east along the trail for 300 m and then north through the woods for 800 m. He can run 160 m per minute along the trail but only 70 m per minute through the woods. Running directly through the woods toward the tree minimizes the distance, but he will be going slowly the whole time. He could instead run 300 m along the trail before entering the woods, maximizing the total distance but minimizing the time in the woods. Perhaps the fastest route is a combination, as shown in Figure 7. Find the path that will get him to the tree in the minimum time. SOLUTION As in Example 1, the first step is to read and understand the problem. If the statement of the problem is not clear to you, go back and reread it until you understand it 7s before moving on. malov word Exer We have already started Step 2 by providing Figure 7. Let x be the distance shown in Figure 7, so the distance he runs on the trail is 300 distance he runs through the woods is V800² + x² . The first part of Step 3 is noting that we are trying to minimize the total amount of time, which is the sum of the time on the trail and the time through the woods. We must express x. By the Pythagorean theorem, the dots this time as a function of x. Since time distance/speed, the total time is 300 x -xU V8002 + x² T(x) = 160 70 to To complete Step 4, notice in this equation that 0 < x < 300. ehasigizemd LOuiddod eviucvitsO grb 14.2 Applications of Extrema 765 0 orp svisvnob zidi We now move to Step 5, in which we find the critical points by calculating the derivative and setting it equal to 0. Since V 8002 + x² =(8002 + x²)2, %3| T'(х) " (х) — (8002 + x²)-1/2 (2x) = 0. 160 70 1 %3D 70V8002 + x² 160 16x = 7V800² + x² Britb, Cross multiplyUT RUOY and divide by 10. Extrema Candidates %3D 256x² = 49(800² + x²) = (49 · 800²) + 49x² 207x = 49 · 800² there х T(x) %3D Square both sides. Subtract 49x from both sides. 0. 13.30 %3D 300 12.21 Minimum 49 8002 x² 207 Wecould Iohbail 7 800 - 389 Take the square root of both sides. %3D V 207 YOUR TURN 2 Suppose the professor in Example 2 can only run 40 m per minute through the woods. Find the path that will get him to the Since 389 is not in the interval [0, 300], the minimum time must occur at one of the endpoints. We now complete Step 6 by creating a table with T(x) evaluated at the endpoints. We see from the table that the time is minimized when x = 300, that is, when the professor heads straight for the tree. %3D tree in the minimum time. TRY YOUR TURN 2