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(c) f (x) = x 3 + 2x 2 + x 1 x2
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(e) f (x) = x 4 2x 3 x 2 4x + 3 (g) f (x =
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1 + on [1, 4] 16 x 1 x 3 x for 0 x 1 3 (i) f (x) = x 2 + x 4 for 1 < x 2 3
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on [ 1, 1] 2x + 5 on [ 5, 3] (f ) f (x) = 2 x 4 3 x (h) f (x) = on [ 1, 1] x+2
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on [0, 2]
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14.8 A farmer wishes to fence in a rectangular eld. If north-south fencing costs $3 per yard, and east west fencing costs $2 per yard, what are the dimensions of the eld of maximum area that can be fenced in for $600 14.9 A farmer has to fence in a rectangular eld alongside a straight-running stream. If the farmer has 120 yards of fencing, and the side of the eld alongside the stream does not have to be fenced, what dimensions of the eld will yield the largest area
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CHAP. 14]
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14.10 The distance by bus from New York to Boston is 225 miles. The bus driver gets paid $12.50 per hour, while the other costs of running the bus at a steady speed of x miles per hour amount to 90 + 0.5x cents per mile. The minimum and maximum legal speeds on the bus route are 40 and 55 miles per hour. At what steady speed should the bus be driven to minimize the total cost 14.11 A charter airline is planning a ight for which it is considering a price of between $150 and $300 per person. The airline estimates that the number of passengers taking the ight will be 200 0.5x, depending on the price of x dollars that will be set. What price will maximize the income 14.12 Suppose that a company can sell x radios per week if it charges 100 0.1x dollars per radio. Its production cost is 30x + 5000 dollars when x radios are produced per week. How many radios should be produced to maximize the pro t, and what will be the selling price per radio 14.13 A box with square base and vertical sides is to be made from 150 square feet of cardboard. What dimensions will provide the greatest volume if: (a) the box has a top surface; (b) the box has an open top 14.14 A farmer wishes to fence in a rectangular eld, and also to divide the eld in half by another fence (AB in Fig. 14-6). The outside fence costs $2 per foot, and the fence in the middle costs $3 per foot. If the farmer has $840 to spend, what dimensions will maximize the total area
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Fig. 14-6
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Fig. 14-7
14.15 On a charter ight the price per passenger is $250 for any number of passengers up to 100. The ight will be canceled if there are fewer than 50 passengers. However, for every passenger over 100, the price per passenger will be decreased by $1. The maximum number of passengers that can be own is 225. What number of passengers will yield the maximum income 14.16 Among all pairs x, y of nonnegative numbers whose sum is 100, nd those pairs: (a) the sum of whose squares x 2 + y2 is a minimum; (b) the sum of whose squares x 2 + y2 is a maximum; (c) the sum of whose cubes x 3 + y3 is a minimum. 14.17 A sports complex is to be built in the form of a rectangular eld with two equal semicircular areas at each end (see Fig. 14-7). If the border of the entire complex is to be a running track 1256 meters long, what should be the dimensions of the complex so that the area of the rectangular eld is a maximum 14.18 A wire of length L is cut into two pieces. The rst piece is bent into a circle and the second piece into a square. Where should the wire be cut so that the total area of the circle plus the square is: (a) a maximum; (b) a minimum geometry The area of a circle of radius r is r 2 , and the circumference is 2 r. 14.19 A wire of length L is cut into two pieces. The rst piece is bent into a square and the second into an equilateral triangle. Where should the wire be cut so that the total area of the square and the triangle is greatest s2 3 . 4
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