Chapter 4 | Applications of Derivatives
443
Since 10+2 7 is not in the domain of consideration, the only critical point we need to consider is 10−2 7. Therefore, the volume is maximized if we let x =10−2 7 in. The maximum volume is V (10 − 2 7) = 640 + 448 7 ≈ 1825 in. 3 as shown in the following graph.
Figure 4.65 Maximizing the volume of the box leads to finding the maximum value of a cubic polynomial.
Watch a video (http://www.openstax.org/l/20_boxvolume) about optimizing the volume of a box.
4.32 Suppose the dimensions of the cardboard in Example 4.33 are 20 in. by 30 in. Let x be the side length of each square and write the volume of the open-top box as a function of x . Determine the domain of consideration for x .
Example 4.34 Minimizing Travel Time
An island is 2mi due north of its closest point along a straight shoreline. A visitor is staying at a cabin on the shore that is 6mi west of that point. The visitor is planning to go from the cabin to the island. Suppose the visitor runs at a rate of 8mph and swims at a rate of 3mph. How far should the visitor run before swimming to minimize the time it takes to reach the island? Solution Step 1: Let x be the distance running and let y be the distance swimming ( Figure 4.66 ). Let T be the time it takes to get from the cabin to the island.
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