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Problems Optimization 1. MAXIMIZING PROFITS The estimated monthly profit ( in dollars) realizable by Cannon Precision Instruments for manufacturing and selling x units of its model M1 camera is P ( x ) = - 0.04 X 2 + 240 x - 10,000 To maximize its profits, how many cameras should Cannon produce each month?
2. LIGHT OF A ROCKET The altitude ( in feet ) attained by a model rocket t sec Into flight is given by the function 1 3 t + 4t 2 + 20t + 2 3 Find the maximum altitude attained by the rocket. h( t ) = 3. FEMALE SELF-EMPLOYED WORKFORCE Data show that the number of nonfarm, full-time, self-employed women can be approximated by N ( t ) = 0.81t - 1.14 t + 1.53 where N ( t ) is measured in millions and t is measured in 5 year intervals, with t = 0 corresponding to the beginning of 1963. Determine the absoluteextrema of the function N on the interval [ 0,6 ] . Interpret your results. 4. AVERAGE SPEED OF A VEHICLE ON A HIGHWAY The average speed of a vehicle on a stretch of Route 134 between 6 A.M. and 10 A.M. on a typical weekday is approximated by the function f ( t ) = 20t - 40 t + 50 where f ( t ) is measured in miles per hour and t is measured in hours, with t = 0 corresponding to 6 A.M. At what time of the morning commute is the traffic moving at the slowest rate ? What is the average speed of a vehicle at that time ?
5. MAXIMIZING PROFITS The management of Trappee and Sons, producers of the famous TexaPep hot sauce, estimate that the profit ( in dollars) from the daily production and sale of x cases
( each case consisting of 24 bottles) of the hot sauce is given by P ( x ) = - 0.000002 x 3 + 6 x - 400
What is the largest possible profit Trappee can make in one day ?
6. MAXIMIZING PROFITS The quantity demanded each month of the Walter Serkin recording of Beethoven' s Moonlight Sonata manufactured by Phonola Record Industries, is related to the price / compact disc. The equation p = - 0.00042x + 6
( 0 x
12,000)
where p denotes the unit price in dollars and x is the number of discs demanded, relates the demand to the price. The total monthly cost ( indollars) for pressing and packaging x copies of this classical recording is given by C ( x ) = 600 + 2t - 0.00002t 2
( 0 x 20,000)
To maximize its profits, how many copies should Phonola produce each month? Hint : The revenue is R ( x ) = px , and the profit is P ( x ) = R ( x ) - C ( x ) .
47. MAXIMIZING PROFIT A manufacturer of tennis rackets finds that the total cost C ( x ) ( in dollars) of manufacturing x rackets / day is given by C ( x ) = 400 + 4 x + 0.0001x 2 Each racket can be sold at a price of p dollars, where p is related to x by the demand equation p = 10 - 0.0004 x If all rackets that are manufactured can be sold, find the daily level of production that will yield a maximum profit for the manufacturer.
48. MAXIMIZING PROFIT The weekly demand for the Pulsar 25 in. color console television is given by the demand equation p = - 0.05 x + 600 (0 x 12,000) where p denotes the wholesale unit price in dollars and x denotes the quantity demanded. The weekly total cost function associated with manufacturing these sets is given by C ( x ) = 0.000002 x 3 - 0.03 x 2 + 400 x + 80,000 where C ( x ) denotes the total cost incurred in producing x sets. Find the level of production that will yield a maximum profit for the manufacturer. Hint : Use the quadratic formula.
49. MAXIMIZING PROFIT A division of Chapman Corporation manufactures a pager. The weekly fixed cost for the division is $20,000, and the variable cost for producing x pagers / week is V ( x ) = 0.000001x 3 - 0.01X 2 + 50 x dollars.The company realizes a revenue of R ( x ) = - 0.02t 2 + 150 x (0 x 7500) dollars from the sale of x pagers / week. Find the level of production that will yield a maximum profit for the manufacturer. Hint : Use the quadratic formula. 50. MINIMIZING AVERAGE COST Suppose the total cost function for manufacturing a certain product is C ( x ) = 0.2 ( 0.01x 2 + 120) dollars, where x represents the number of units produced. Find the level of production that will minimize the average cost. 51. MINIMIZING PRODUCTION COSTS The total monthly cost ( in dollars) incurred by Cannon Precision Instruments for manufacturing x units of the model M1 camera is given by the function C ( x ) = 0.0025 x 2 + 80 x + 10,000 a) Find the average cost function C. b) Find the level of production that results in the smallest average production cost. c) Find the level of production for which the average cost is equal to the marginal cost. d) Compare the result of part ( c) with that of part ( b) .
52. MINIMIZING PRODUCTION COSTS The daily total cost ( in dollars) incurred by Trappee and Sons for producing x cases of TexaPep hot sauce is given by the function C ( x ) = 0.000002 x 3 + 5 x + 400 a) Find the average cost function C. b) Find the level of production that results in the smallest average production cost. c) Find the level of production for which the average cost is equal to the marginal cost. d) Compare the result of part ( c) with that of part ( b) .
53. MAXIMIZING REVENUE Suppose the quantity demanded per week of a certain dress is related to the unit price p by the demand equation p = 800 - x , where p is in dollars and x is the number of dresses made. To maximize the revenue, how many dresses should be made and sold each week ? Hint : R ( x ) = px .
54. MAXIMIZING REVENUE The quantity demanded each month of the Sicard wristwatch is related to the unit price by the equation 50 p= ( 0 x 20) 0.01x 2 + 1 where p is measured in dollars and x is measured in units of a thousand. To yield a maximum revenue, how many watches must be sold?
55. OXYGEN CONTENT OF A POND When organic waste is dumped into a pond, the oxidation process that takes place reduces the pond' s oxygen content. However, given time,nature will restore the oxygen content to its natural level. Suppose the oxygen content t days after organic waste has been dumped into the pond is given by - 4t + 4 t2 f ( t ) = 100 2 (0 t ) t + 4 percent of its normal level. a) When is the level of oxygen content lowest ? b) When is the rate of oxygen regeneration greatest ?
56. AIR POLLUTION The amount of nitrogen dioxide, a brown gas that impairs breathing, present in the atmosphere on a certain May day in the city of Long Beach is approximated by 136 A( t ) = + 28 ( 0 t P 11) 2 1+ 0.25 ( t - 4.5) where A ( t ) is measured in pollutant standard index ( PSI) and t is measured in hours, with t = 0 corresponding to 7 AM. Determine the time of day when the pollution is at its highest level.
57. MAXIMIZING REVENUE The average revenue is defined by the function R( x) ( x > 0) x Prove that if a revenue function R ( x ) is concave downward '' ( x ) < 0 R , then the level of sales that will result in the largest average revenuw occurs when R ( x ) = R ' ( x ) . R( x) = 58. VELOCITY OF BLOOD According to a law discovered by the 19th century physician Jean Louis Marie Poiseuille, the velocity ( in centimeters / second) of blood r cm. from the central axis of an artery is given by v ( r ) = k (R 2 - r 2 ) where k is a constant and R is the radius of the artery. Show that the velocity of blood is greatest along the central axis.
59. GDP OF A DEVELOPING COUNTRY A developing country's gross domestic product ( GDP) from 1993 to 2001 is approximated by the function G ( t ) = - 0.2t 3 + 2.4t 2 + 60 rate of the country's GDP was maximal in 1997.
( 0 t x 8)
where G ( t ) is measured in billions of dollars and t = 0 corresponds to 1993. Show that the growth
60. CRIME RATES The number of major crimes committed in the city of Bronxville between 1987 and 1994 is approximated by the function N ( t ) = - 0.1t 3 + 1.5t 2 + 100 (0 x t 7) where N ( t ) denotes the number of crimes committed in year t (t = 0 corresponds to 1987). Enraged by the dramatic increase in the crime rate, the citizens of Bronxville, with the help of the local police, organized "Neighborhood Crime Watch"groups in early 1991 to combat this menace. Show that the growth in the crime rate was maximal in 1992, giving credence to the claim that the Neighborhood Crime Watch program was working.
61. SOCIAL SECURITY SURPLUS Data show that the estimated cash in the Social Security retirement and disability trust funds may be approximated by f ( t ) = - 0.0129t 4 + 0.3087t 3 + 2.1760t 2 + 62.8466t + 506.2955 where f ( t ) is measured in billions of dollars and t is measured in years, with t = 0 corresponding to 1995. Show that the Social Security surplus willbe at its highest level at approximately the middle of the year 2018. Hint : Show that t = 23.6811 is an approximate critical point of f ' ( t ) .
62. ENERGY EXPENDED BY A FISH It has been conjectured that a fish swimming a distance of L ft at a speed of v ft / sec relative to the water and against a current flowing at the rateof u ft / sec (u < v ) expends a total energy given by aLv 3 v- u where E is measured in foot-pounds ( ft-Ib) and a is a constant. Find the speed v at which E(v) = the fish must swim in order to minimize the total energy expended. (Note : This result has been verified by biologists.)
63. REACTION TO A DRUG The strength of a human body ' s reaction R to a dosage D of a certain drug is given by D k R = D2 3 2 where k is a positive constant. Show that the maximum reaction is achieved if the dosage is k units. 64. Refer to the prior exercise. Show that the rate ofchange in the reaction R with respect to the dosage D is maximal if D = kl2.
1. ENCLOSING THE LARGEST AREA The owner of the Rancho Los Feliz has 3000 yd of fencing material with which to enclose a rectangular piece of grazing land along the straight portion of a river. If fencing is not required along the river, what are the dimensions of the largest area that he can enclose ? What is this area ?
2. ENCLOSING THE LARGEST AREA Refer to the prior exercise. As an alternative plan, the owner of the Rancho Los Feliz might use the 3000 yd of fencing material to enclose the rectangular piece of grazing land along the straight portion of the river and then subdivide it by means of a fence running parallel to the sides. Again, no fencing is required along the river. What are the dimensions of the largest area that can be enclosed? What is this area ? ( See the accompanying MINIMIZING figure.)
1. CONSTRUCTION COSTS The management of the UNICO department store has decided to enclose an 800 ft 2 area outside the building for displaying potted plants and flowers. One side will be formed by the external wall of the store, two sides will be constructed of pineboards, and the fourth side will be made of galvanized steel fencing material. If the pine board fencing costs $6 / running foot and the steel fencing costs $3 / running foot, determine the dimensions of the enclosure that can be erected at minimum cost.
2. PACKAGING By cutting away identical squares from each corner of a rectangular piece of cardboard and folding up the resulting flaps, an open box may be made. If the cardboard is 15 in. long and 8 in. wide, find the dimensions of the box that will yield the maximum volume. 5. METAL FABRICATION If an open box is made from a tin sheet 8 in. square by cutting out identical squares from each corner and bending up the resulting flaps, determine the dimensions of the largest box that can be made. 6. MINIMIZING PACKAGING COSTS If an open box has a square base and a volume of 108 in.3 and is constructed from a tin sheet, find the dimensions of the box, assuming a minimum amount of material is used in its construction. 7. MINIMIZING PACKAGING COSTS What are the dimensions of a closed rectangular box that has a square cross section, a capacity of 128 in.3 , and is constructed using the least amount of material?
8. MINIMIZING PACKAGING COSTS A rectangular box is to have a square base and a volume of 20 ft 3 . If the material for the base costs 30 / square foot, the material for the sides costs 10 / square foot, and the material for the top costs 20 / square foot, determine the dimensions of the box that can be constructed at minimum cost.
9. PARCEL POST REGULATIONS Postal regulations specify that a parcel sent by parcel post may have a combined length and girth of no more than 108 in. Find the dimensions of a rectangular package that has a square cross section and the largest volume that may be sent through the mail. What is the volume of such a package ? Hint : The length plus the girth is 4 x + h ( see the accompanying figure) .
10. BOOK DESIGN A book designer has decided that the pages of a book should have 1 in. margins at the top and 1 bottom and in. margins on the sides. She further stipulated that each page should have 2 an area of 50 sq in? ( see the accompanying figure) . Determine the page dimensions that will result in the maximum printed area on the page.
11. PARCEL POST REGULATIONS Postal regulations specify that a parcel sent by parcel post may have a combined length and girth of no more than 108 in. Find the dimensions of the cylindrical package of greatest volume that may be sent through the mail. What is the volume of such a package ?
12. MINIMIZING COSTS For its beef stew, Betty Moore Company uses aluminum containers that have the form of right circular cylinders. Find the radius and height of a container if it has a capacity of 36 and is constructed using the least amount of metal.
13. PRODUCT DESIGN The cabinet that will enclose the Acrosonic model D loudspeaker will be rectangular and will have an internal volume of 2.4 cubic feet. For aesthetic reasons, it has been decided that the height of the cabinet is 1.5 times its width. If the top, bottom,and sides of the cabinet are constructed of veneer costing 40 cents / square foot and the front
( ignore the cutouts in the baffle) and rear are constructed of particle board costing
20 cents / square foot, what are the dimensions of the enclosure that can be constructed at a minimum cost ? 13. DESIGNING A NORMAN WINDOW A Norman window has the shape of a rectangle surmounted by a semicircle. If a Norman window is to have a perimeter of 28 ft, what should its dimensions be inorder to allow the maximum amount of light through the window ?
15. OPTIMAL CHARTER FLIGHTFARE If exactly 200 people sign up for a charter flight, Leisure World Travel Agency charges $300 / person. However, if more than 200 people sign up for the flight ( assume this is the case) , then each fare is reduced by $1 for each additional person. Determine how many passengers will result in a maximum revenue for the travel agency? What is the maximum revenue ? What would be the fare per passenger in this case ? Hint : Let x denote the number of passengers above 200. Show that the revenue function R is given by R ( x ) = ( 200 + x ) (300 - x ). 16. MAXIMIZING YIELD An apple orchard has an average yield of 36 bushels of apples / tree if tree density is 22 trees / acre. For each unit increase in tree density, the yield decreases by 2 bushels. How many trees should be planted in order to maximize the yield?
17. CHARTER REVENUE The owner of a luxury motor yacht that sails among the 4000 Greek islands charges $600 / person / day if exactly 20 people sign up for the cruise. However, if more than 20 people sign up (up to the maximum capacity of 90) for the cruise, theneach fare is reduced by $4 for each additional passenger. Assuming at least 20 people sign up for the cruise, determine how many passengers will result in the maximum revenue for the owner of the yacht. What is the maximum revenue ? What would be the fare / passenger in this case ? 18. PROFIT OF A VINEYARD Phillip, the proprietor of a vineyard, estimates that the rust 10,000 bottles of wine produced this season will fetch a profit of $5 / bottle. However, the profit from each bottle beyond 10,000 drops by $0.0002 for each additional bottle sold. Assuming at least 10,000 bottles of wine are produced and sold, what is the maximum profit ? What would be the price / bottle in this case ?
19. STRENGTH OF A WOODEN BEAM A wooden beam has a rectangular cross-section of height h in, and width w in. (see the accompanying figure). The strength S of the beam is directly proportional to its width and the square of its height. What are the dimensions of the cross section of the strongest beam that can be cut from a round log of diameter 24 in.? Hint : S = kh 2w , where k is a constant of proportionality.
20. DESIGNING A GRAIN SILO A grain silo has the shape of a right circular cylinder surmounted by a hemisphere
( see the accompanying figure) . If the silo is to have a capacity of 504p ft 3 , find the
radius and height of the silo that require the least amount of material to construct. 2 Hint : The volume of the silo is pr 2 h + pr 3 , and the surface area ( including the floor ) 3 2 is p(3r + 2rh).
21. MINIMIZING COST OF LAYING CABLE In the following diagram, S represents the position of a power relay station located on a straight coast,and E shows the location of a marine biology experimental station on an island. A cable is to be laid connecting the relay station with the experimental station. If the cost of running the cable on land is $1.50 / running foot and the cost of running the cable under water is $2.50 / running foot, locate the point P that will result in a minimum cost (solve for x ).
22. STORING RADIOACTIVE WASTE A cylindrical container for storing radioactive waste is to be constructed from lead and have a thickness of 6 in. ( see the accompanying figure ) . If the volume of the outside cylinder is to be 16pft 3 , find the radius and the height of the inside cylinder that will result in a container of maximum storage capacity.
23. FLIGHTS OF BIRDS During daylight hours, some birds fly more slowly over water than over land because some of their energy is expended in overcoming the downdrafts of air over open bodies of water. Suppose a bird that flies at a constant speed of 4 mph over water and 6 mph over land starts its journey at the point E on an island on the shore of the mainland,as shown in the accompanying figure, Find the location of the point P that allows the bird to complete its journey in the minimum time (solve for x ).
24. OPTIMAL SPEED OF A TRUCK 400 A truck gets mpg when driven at a constant speed of x mph ( between 50 and 70 mph) . x If the price of fuel is $1/ gallon and the driver is paid $8 / hour, at what speed between 50 and 70 mph is it most economical to drive ?
25. INVENTORY CONTROL AND PLANNING The demand for motorcycle tires imported by Dixie Import - Export is 40,000 / year and may be assumed to be uniform throughout the year. The cost of ordering a shipment of tires is $400, and the cost of storing each tire for a year is $2. Determine how many tires should be in each shipment if the ordering and storage costs are to be minimized.
( Assume that each shipment arrives just as the previous one has been sold.)
26. INVENTORY CONTROL AND PLANNING McDuff Preserves expects to bottle and sell 2,000,000 32-oz jars of jam. The company orders its containers from Consolidated Bottle Company. The cost of ordering a shipment of bottles is $200 and the cost of storing each empty bottle for a year is $0.44 How many orders should McDuff place per year and how many bottles should be in each shipment if the ordering and storage costs are to be minimized?
( Assume that each shipment of bottles is used up before the next shipment arrives.)
27. INVENTORY CONTROL AND PLANNING Neilsen Cookie Company sells its assorted butter cookies in containers that have a net content of 1 lb. The estimated demand for the cookies is 1,000,000 1-lb containers. The setup cost for each production run is $500, and the manufacturing cost is $0.50 for each container of cookies. The cost of storing each container of cookies over the year is $0.40. Assuming uniformity of demand throughout the year and instantaneous production of how many containers of cookies should Neilsen produce per production run in order to minimize the production cost ? Hint : Show that the total production cost is given by 500,000,000 C( x) = + 0.2 x + 500,000 x Then minimize the function C on the interval ( 0, 1,000,000) .
28. RACETRACK DESIGN The accompanying figure depicts a racetrack with ends that are semicircular inshape. 1 The length of the track is 1760 ft miFind l and r so that the area enclosed by the . 3 rectangular region of the racetrack is as large as possible. What is the area enclosed by the track in this case ?
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