Applications of Functions



Applications of Functions

1. Two numbers add to 5. What is the largest possible value of their product?

2. Find two numbers adding to 20 such that the sum of their squares is as small as possible.

3. The difference of two numbers is 1. What is the smallest possible value for the sum of their squares?

4. For each quadratic function specified below, state whether it would make sense to look for a highest or a lowest point on the graph. Then determine the coordinates of that point.

a. y = 2x2 – 8x + 1

b. y = –3x2 – 4x – 9

c. h = –16t2 + 256t

d. f(x) = 1 – (x+1)2

e. g(t) = t2 + 1

f. f(x) = 1000x2 – x + 100

5. Among all rectangles having a perimeter of 25 m, find the dimensions of the one with the largest area.

6. What is the largest possible area for a rectangle whose perimeter is 80 cm?

7. What is the largest possible area for a for a right triangle in which the sum of the lengths of the two shorter sides is 100 in?

8. The perimeter of a rectangle is 12 m. Find the dimensions for which the diagonal is as short as possible.

9. a) Minimize S = 6x2 – 2xy + 5y2 given that x + y = 13.

b) Minimize S = 12x2 + 4xy – 10y2 given that x + y = 14.

c) Maximize S = 3x2 + 5xy – 2y2 given that x + y = 8.

d) Maximize S = 4x2 + 3xy – 5y2 given that x + y = –8.

e) Maximize S = –2x2 + 3xy – 5y2 given that x + y = 20.

f) Minimize S = 3x2 + 2xy + 2y2 given that 3x – 2y = 42.

g) Minimize S = 2x2 + 5xy – y2 given that 4x – y = 12.

h) Minimize S = 3x2 – xy + 2y2 given that x – 2y = 24.

i) Maximize S = –3x2 + 2xy + 4y2 given that 2x – 5y = –9.

j) Maximize S = –x2 + 6xy – 7y2 given that 2x – 3y = 2.

10. Two numbers add to 6.

a. Let T denote the sum of the squares of the two numbers. What is the smallest possible value for T?

b. Let S denote the sum of the first number and the square of the second. What is the smallest possible value for S?

c. Let U denote the sum of the first number and twice the square of the second number. What is the smallest possible value for U?

d. Let V denote the sum of the first number and the square of twice the second. What is the smallest possible value for V?

11. Suppose that the height of an object shot straight up is given by (h in feet and t in seconds). Find the maximum height and the time at which the object hits the ground.

12. A baseball is thrown straight up, and its height as a function of time is given by the formula h(t) = –16t2 + 32t (h in feet and t in seconds).

a. Find the height of the ball when t = 1 and when t = .

b. Find the maximum height of the ball and the time at which that height is attained.

c. At what time(s) is the height 7 feet?

13. What point is nearest to (3, 0) on the curve y = ?

14. What point is closest to (4, 1) on the curve y = ?

15. Find the coordinates of the point on the line y = 3x + 1 closest to (4, 0).

16. a. What number exceeds its square by the greatest amount?

b. What number exceeds twice its square by the greatest amount?

17. Suppose that you have 1800 meters of fencing available with which to build three adjacent rectangular corrals as shown in the figure. Find the dimensions so that the total enclosed area is as large as possible.

18. Five hundred feet of fencing is available for a rectangular pasture alongside a river, the river serving as one side of the rectangle (so only three sides require fencing). Find the dimensions yielding the greatest area.

19. Let A = 3x2 + 4x – 5 and B = x2 – 4x – 1. Find the minimum value of A – B.

20. Let R = 0.4x2 + 10x + 5 and C = 0.5x2 + 2x + 101. For which value of x is R – C a maximum?

21. Suppose that the revenue generated by selling x units of a certain commodity is given by R = – x2 + 200x. Assume that R is in dollars. What is the maximum revenue possible in this situation?

22. Suppose that the function p(x) = – x + 30 relates the selling price p of an item to the quantity sold, x. Assume P is in dollars. For which value of x will the corresponding revenue be a maximum? What is this maximum revenue and what is the unit price in this case?

23. A piece of wire 200 cm long is to be cut into two pieces of lengths x and 200 – x. The first piece is to be bent into a circle and the second piece into a square. For which value of x is the combined area of the circle and square as small as possible?

24. A 30 in piece of string is to be cut into two pieces. The first piece will be formed into the shape of an equilateral triangle and the second piece into a square. Find the length of the first piece if the combined area of the triangle and the square is to be as small as possible?

25. a. Same as exercise 23 except both pieces are to be formed into squares.

b. Could you have guessed the answer to part a?

26. The action of sunlight on automobile exhaust produces air pollutants known as photochemical oxidants. In a study of cross-country runners in Los Angeles, it was shown that running performances can be adversely affected when the oxidant level reaches 0.03 parts per million. Let us suppose that on a given day the oxidant level L is approximated by the formula

L = 0.059t2 – 0.354t + 0.557 (0 t 7).

Here, t is measured in hours, with t = 0 corresponding to 12 noon, and L is in parts per million. At what time is the oxidant level L a minimum? At this time, is the oxidant level high enough to affect a runner's performance?

27. If x + y = 1, find the largest possible value of the quantity x2 – 2y2.

28. a. Find the smallest possible value of the quantity x2 + y2 under the restriction that 2x + 3y = 6.

b. Find the radius of the circle whose center is at the origin and that is tangent to the line 2x + 3y = 6. How does this answer relate to your answer in part a?

29. Through a type of chemical reaction known as autocatalysys, the human body produces the enzyme trypsin from the enzyme trypsinogen. (Trypsin then breaks down proteins into amino acids, which the body needs for growth.) Let r denote the rate of this chemical reaction in which trypsin is formed from trypsinogen. It has been shown experimentally that r = kx(a – x), where r is the rate of the reaction, k is a positive constant, a is the initial amount of trypsinogen, and x is the amount of trypsin produced (so x increases as the reaction proceeds). Show that the reaction rate r is a maximum when x = . In other words, the speed of the reaction is the greatest when the amount of trypsin formed is half of the original amount of trypsinogen.

30. a. Let x + y = 15. Find the minimum value of the quantity x2 + y2.

b. Let C be a constant and x + y = C. Show that the minimum value of x2 + y2 is . Then use this result to check your answer in part a.

31. Suppose that A, B, and C are positive constants and that x + y = C. Show that the minimum value of Ax2 + By2 occurs when x = and y = .

32. The figure at the right shows two concentric squares. For which value of x is the shaded area a maximum?

33. Find the largest value of the function f(x) = .

34. A rancher, who wishes to fence off a rectangular area, finds that the fencing in the east-west direction will require extra reinforcement due to the strong prevailing winds. Because of this, the cost of fencing in the east-west direction will be $12 per linear yard, as opposed to a cost of $8 per yard for fencing in the north-south direction. Find the dimensions of the largest possible rectangular area that can be fenced for $4800.

35. Let f(x) = (x – a)2 + (x – b)2 + (x – c)2, where a, b, and c are constants. Show that f(x) will be a minimum when x is the average of a, b, and c.

36. Let y = a1(x – x1)2 + a2(x – x2)2, where a1, a2, x1, and x2 are all constants. Further, suppose that a1 and a2 are both positive. Show that the minimum of this function occurs when x = .

37. Among all rectangles with a given perimeter P, find the dimensions of the one with the shortest diagonal.

38. The figure at the right shows a rectangle inscribed in a given triangle of base b and height h. Find the ratio of the area of the triangle to the area of the rectangle when the area of the rectangle is a maximum.

39. a. Find the coordinates of the point on the line y = mx + b that is closest to the origin.

b. Find the perpendicular distance from the origin to the line y = mx + b.

c. Use part b. to find the perpendicular distance from the origin to the line Ax + By + C = 0.

40. The point P lies in the first quadrant on the graph of the line y = 7 – 3x. From the point P, perpendiculars are drawn to the x axis and the y axis, respectively. What is the largest possible area for the rectangle thus formed?

41. Find the largest possible area for the shaded rectangle shown in the figure is Then use this to check your answer in exercise 39.

42. Show that the maximum possible area for a rectangle inscribed in a circle of radius R is 2R2.

43. A Norman window is in the shape of a rectangle surmounted by a semicircle, as shown in the figure. Assume that the perimeter of the window is P, a constant. Find the values of h and r when the area is a maximum and find this area.

44. A rectangle is inscribed in a semicircle of radius 3. Find the largest possible area for this rectangle.

45. An athletic field with a perimeter of mi consists of a rectangle with a semicircle at each end, as shown in the figure below. Find the dimensions x and r that yield the greatest possible area for the rectangular region.

46. Find the values of x that make y a minimum or a maximum, as the case may be. Find the corresponding y value and indicate whether it is a minimum or a maximum.

a. y = 3x4 – 12x2 – 5

b. y =

47. By analyzing sales figures, the economist for a stereo manufacturer knows that 150 units of a top of the line turntable can be sold each month when the price is set at p = $200 per unit. The figures also show that for each $10 hike in price, 5 fewer units are sold each month.

a. Let x denote the number of units sold per month and let p denote the price per unit. Find a linear function relating p and x.

b. Express the revenue R as a function of x.

c. What is the maximum revenue? At what level should the price be set to achieve this maximum revenue?

48. Imagine that you own an orchard of orange trees. Suppose from past experience you know that when 100 trees are planted, each tree will yield approximately 240 oranges. Furthermore, you've noticed that when additional trees are planted, the yield per (each) tree in the orchard decreases. Specifically, you have noted that the yield per tree decreases by about 20 oranges for each additional tree planted. Approximately how many trees should be planted in the orchard to produce the largest possible total yield of oranges?

49. An appliance firm is marketing a new refrigerator. It determines that in order to sell x refrigerators, its price per refrigerator must be p = D(x) = 280 – 0.4x. It also determines that its total cost of producing x refrigerators is given by .

a) How many refrigerators must the company produce and sell in order to maximize profit?

b) What is the maximum profit?

c) What price per refrigerator must be charged in order to make this maximum profit?

50. The owner of a 30 unit motel find that all units are occupied when the charge is $20 per day per unit. For every increase of x dollars in the daily rate, there are x units vacant. Each occupied room costs $2 per day to service and maintain. What should he charge per unit per day in order to maximize profit?

51. A university is trying to determine what price to charge for football tickets. At a price of $6 per ticket, it averages $70,000 people per game. For every increase of $1, it loses 10,000 people from the average number. Every person at the game spends an average of $1.50 on concessions. What price per ticket should be charged in order to maximize revenue? How many people will attend at that price?

52. When a theater owner charges $3 for admission, there is an average attendance of 100 people. For every 10 cent increase in admission, there is a loss of 1 customer from the average. What admission should be charged in order to maximize revenue?

53. An apple farm yields an average of 30 bushels of apples per tree when 20 trees are planted on an acre of ground. Each time 1 more tree is planted per acre, the yield decreases 1 bushel per tree due to the extra congestion. How many trees should be planted in order to get the highest yield?

54. A triangle is removed from a semicircle of radius R as shown in the figure. Find the area of the remaining portion of the circle if it is to be a minimum.

55. Let f(x) = x2 + px + q, and suppose that the minimum value of this function is 0. Show that q = .

56. Suppose that x and y are both positive numbers and that their sum is 4. Find the smallest possible value for the quantity .

Answers

1. 6.25

3.

5. 6.25 × 6.25

7. 1250 in2

9. a) x = 6, y = 7, S = 377 b) x = 4, y = 10, S = 952

c) x = 9, y = –1, S = 196 d) x = –13, y = 5, S = 356

e) x = 13, y = 7, S = –310 f) x = 8, y = –9 S = 210

g) x = –3, y = –24, S = –198 h) x = 2, y = –11, S = 276

i) x = 3, y = 3, S = 27 j) x = 10, y = 6, S = 8

10. a. 18; b. ; c. ; d.

11. a. 16 ft, 12 ft

b. 16 ft at 1 sec

c. sec, sec

14. ,

16. a. ; b.

18. 125 (2 sides) × 250 ft

20. 40

22. 60; $900, $15

24. approximately 21.7

26. 3 PM; no

28. a.

30. a.

32.

34. 100 × 150

38. 2

40. square units

41. 2R2

44. 9 square units

46. a. (±, –17)

b. min at (0, 0) and (4, 0); max at (2, 2)

48. 56 trees

50. $26

52. $6.50

54. R2

-----------------------

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download