Download CHAPTER 12 Quadratic Functions 102

102
CHAPTER 12 Quadratic Functions
12.14. Find two real numbers whose sum is S and whose product is a maximum.
Let one number be x; then the other number must be S ! x. Then the product is a quadratic function of x:
P(x) " x(S ! x) " !x2 # Sx. By completing the square, this function can be written as P(x) " !(x ! S/2)2 # S2/4.
Thus the maximum value of the function occurs when x " S/2. The two numbers are both S/2.
12.15. A salesperson finds that if he visits 20 stores per week, average sales are 30 units per store each week;
however, for each additional store that he visits per week, sales decrease by 1 unit. How many stores should
he visit each week to maximize overall sales?
Let x represent the number of additional stores. Then the number of visits is given by 20 # x and the corresponding
sales are 30 ! x per store. Total sales are then given by S(x) " (30 ! x)(20 # x) " 600 # 10x ! x2. This is a
quadratic function. Completing the square gives S(x) " !(x ! 5)2 # 625. This has a maximum value when
x " 5; thus, the salesperson should visit 5 additional stores, a total of 25 stores, to maximize overall sales.
SUPPLEMENTARY PROBLEMS
12.16. Show that, for negative a, the quadratic function f (x) " a (x !h)2 # k has a maximum value of k, attained
at x " h.
12.17. Find the maximum or minimum value and graph the quadratic function f(x) " x2 # 6x # 9.
Ans.
Minimum value: 0 when x " !3. (See Fig. 12-13.)
Figure 12-13
12.18. Find the maximum or minimum value and graph the quadratic function f(x) " 6x2 ! 15x.
Ans.
Minimum value: !
5
75
when x " . (See Fig. 12-14.)
8
4
Figure 12-14
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4
12.19. Find the maximum or minimum value and graph the quadratic function f (x) 5 ! x2 ! x # 6.
2
3
Ans.
Maximum value:
170
4
when x " ! . (See Fig. 12-15.)
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9
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CHAPTER 12 Quadratic Functions
Figure 12-15
12.20. State the domain and range for each quadratic function:
1
(a) f (x) ! 3(x " 2)2 # 5; (b) f (x) ! " (x # 3)2 " 7; (c) f (x) ! 6 " x2; (d) f (x) ! x2 " 8x
2
Ans. (a) domain: R, range [5,`); (b) domain: R, range ("`,"7];
(c) domain: R, range ("`,6]; (d) domain: R, range ["16,`)
12.21. A projectile is thrown up from an initial height of 72 feet with an initial velocity of 160 ft/sec2. Its height h(t)
at time t is given by h(t) ! "16t2 # 160t # 72. Find its maximum height, the time when this maximum
height is reached, and the time when the projectile hits the ground.
Ans. Maximum height: 472 feet. Time of maximum height: 5 seconds.
Projectile hits ground: 5 # 2118>2 < 10.4 seconds.
12.22. 1500 feet of chain link fence are to be used to construct six animal cages as in Fig. 12-16.
Figure 12-16
Express the total enclosed area as a function of the width x. Find the maximum value of this area and the
dimensions that yield this area.
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Ans. Area: A(x) ! x (1500 " 3x). Maximum value: 46,875 square feet. Dimensions: 250 feet by 187.5 feet.
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12.23. Find two real numbers whose difference is S and whose product is a minimum.
Ans. S/2 and "S/2
12.24. A basketball team finds that if it charges $25 per ticket, the average attendance per game is 400. For each $.50
decrease in the price per ticket, attendance increases by 10. What ticket price yields the maximum revenue?
Ans.
$22.50