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Unformatted text preview: Contents 8 Applications of Maxima and Minima 127 8.1 The Use of Auxiliary Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 2 CONTENTS Chapter 8 Applications of Maxima and Minima Prerequisites When you write a solution, it is your responsibility to inform the reader what you are doing. You may assume the marker is intelligent but not a mind reader. Therefore you must invest some time in the organization and documentation of your work. Study the following problems and solution with these points in mind. Problem A lighthouse is at point P , 3 miles offshore, from the nearest point 0 of a straight beach. A store is located 5 miles down the beach from 0. The lighthouse keeper can row 3.25 miles/hour and walk 4 miles/hour. How far along the beach from 0 should he land in order to get to the store in the minimum time? Solution: # # # # # # # # # # # # # 5 x 3 √ x 2 + 9 x X P S Let x be the distance from 0 to the point of landing X and let S be the location of the store 5 miles from 0. A. Compute the time function for the trip: The time required to row from P to X is √ x 2 +9 3 . 25 . (Recall that velocity = distance/time). The time required to walk from X to S is 5 x 4 . The total time, T , for the trip is given by: T ( x ) = √ x 2 + 9 3 . 25 + 5 x 4 . In this problem it only makes sense for x to vary from 0 to 5 miles. Therefore we need to find the minimum of T on the closed interval [0 , 5]. 128 Applications of Maxima and Minima B. Compute the critical points of the function: dT dx = x 3 . 25 √ x 2 + 9 1 4 . Therefore dT dx = ⇔ 4 x = 3 . 25 p x 2 + 9 ⇔ 16 x 2 = (3 . 25) 2 ( x 2 + 9) ⇔ { 16 (3 . 25) 2 } x 2 = 9(3 . 25) 2 ⇔ x 2 ∼ = 17 . 482759 ⇔ x ∼ = 4 . 1812389 or 4 . 1812389 C. Evaluate the function at the critical and endpoints to find the value of x that minimizes the trip. T (0) ∼ = 2 . 1730769 T (5) ∼ = 1 . 794139 T (4 . 1812389) ∼ = 1 . 788118 D. Conclusion. Choose x = 4 . 1812389 to minimize the time of the trip. Observe that a very slight increase in rowing speed would make rowing directly to S the fastest route. The following problems may be done by using a four step technique which is outlined below. Step 1 (a) Draw a diagram. The diagram should be large and neatly drawn, occupying about 1 3 page. (b) Label fixed quantities with numerical values. Label the dimensions that vary with letters. Choose letters for any other items that are involved in the problem, e.g. volume. Step 2 (a) Select the quantity that is to be made maximum or minimum. Express it as a function of other quantities. (b) If you are to maximize something which is a function of two variables, then you must find a relationship between the two variables. Usually the geometry of the situation gives a relation. (e.g. similar triangles, Pythagoras theorem.) (c) Express the quantity to be maximized as a function of only one variable....
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This note was uploaded on 09/22/2011 for the course ECON 101 taught by Professor Mr.tull during the Spring '11 term at De La Salle University.
 Spring '11
 Mr.Tull
 Microeconomics

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