121616949-math.127 - A to the closest point C on the road...

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6.1 Optimization 113 We want to know the minimum value of this function when r is in (0 , ). We now set 0 = f ( r ) = - 2 cV/r 2 + 4 Ncπr , giving r = 3 p V/ (2 ). Since f ( r ) = 4 cV/r 3 + 4 Ncπ is positive when r is positive, there is a local minimum at the critical value, and hence a global minimum since there is only one critical value. Finally, since h = V/ ( πr 2 ), h r = V πr 3 = V π ( V/ (2 )) = 2 N, so the minimum cost occurs when the height h is 2 N times the radius. If, for example, there is no difference in the cost of materials, the height is twice the radius (or the height is equal to the diameter). . x a - x A D B C b Figure 6.5 Minimizing travel time. EXAMPLE 6.12 Suppose you want to reach a point A that is located across the sand from a nearby road (see figure 6.5 ). Suppose that the road is straight, and b is the distance from A to the closest point
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Unformatted text preview: A to the closest point C on the road. Let v be your speed on the road, and let w , which is less than v , be your speed on the sand. Right now you are at the point D , which is a distance a from C . At what point B should you turn off the road and head across the sand in order to minimize your travel time to A ? Let x be the distance short of C where you turn off, i.e., the distance from B to C . We want to minimize the total travel time. Recall that when traveling at constant velocity, time=distance/velocity. You travel the distance DB at speed v , and then the distance BA at speed w . Since DB = a-x and, by the Pythagorean theorem, BA = √ x 2 + b 2 , the total time for the trip is f ( x ) = a-x v + √ x 2 + b 2 w ....
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