SETION 4.7 OTIMIZTION ROLEMS 331 and Ris called te revenue function. Te derivative R of te revenue function is called te marginal revenue function and is te rate of cange of revenue wit respect to te numer of units sold. If x units are sold, ten te total profit is and is called te profit function. Te marginal profit function is, te derivative of te profit function. In Exercises 57 6 you are asked to use te marginal cost, revenue, and profit functions to minimize costs and maximize revenues and profits. v EXMLE 6 store as een selling 00 lu-ray disc players a week at $350 eac. market survey indicates tat for eac $10 reate offered to uyers, te numer of units sold will increase y 0 a week. Find te demand function and te revenue function. How large a reate sould te store offer to maximize its revenue? SOLUTION If x is te numer of lu-ray players sold per week, ten te weekly increase in sales is x 00. For eac increase of 0 units sold, te price is decreased y $10. So for 1 eac additional unit sold, te decrease in price will e 0 10 and te demand function is Te revenue function is x R x x p x 350 10 0 x 00 450 1 x R x xp x 450x 1 x Since R x 450 x, we see tat R x 0 wen x 450. Tis value of x gives an asolute maximum y te First Derivative Test (or simply y oserving tat te grap of R is a paraola tat opens downward). Te corresponding price is p 450 450 1 450 5 and te reate is 350 5 15. Terefore, to maximize revenue, te store sould offer a reate of $15. 4.7 Exercises 1. onsider te following prolem: Find two numers wose sum is 3 and wose product is a maximum. (a) Make a tale of values, like te following one, so tat te sum of te numers in te first two columns is always 3. On te asis of te evidence in your tale, estimate te answer to te prolem. First numer Second numer roduct 1 1 4 3 0 60......... () Use calculus to solve te prolem and compare wit your answer to part (a).. Find two numers wose difference is 100 and wose product is a minimum. 3. Find two positive numers wose product is 100 and wose sum is a minimum. 4. Te sum of two positive numers is 16. Wat is te smallest possile value of te sum of teir squares? 5. Wat is te maximum vertical distance etween te line y x and te paraola y x for 1 x? 6. Wat is te minimum vertical distance etween te paraolas y x 1 and y x x? ; Graping calculator or computer required S omputer algera system required 1. Homework Hints availale at stewartcalculus.com
33 HTER 4 LITIONS OF DIFFERENTITION 7. Find te dimensions of a rectangle wit perimeter 100 m wose area is as large as possile. 8. Find te dimensions of a rectangle wit area 1000 m wose perimeter is as small as possile. 9. model used for te yield Y of an agricultural crop as a function of te nitrogen level N in te soil (measured in appropriate units) is Y kn 1 N were k is a positive constant. Wat nitrogen level gives te est yield? 10. Te rate in mg caron m 3 at wic potosyntesis takes place for a species of pytoplankton is modeled y te function 100I I I 4 were I is te ligt intensity (measured in tousands of footcandles). For wat ligt intensity is a maximum? 11. onsider te following prolem: farmer wit 750 ft of fencing wants to enclose a rectangular area and ten divide it into four pens wit fencing parallel to one side of te rectangle. Wat is te largest possile total area of te four pens? (a) Draw several diagrams illustrating te situation, some wit sallow, wide pens and some wit deep, narrow pens. Find te total areas of tese configurations. Does it appear tat tere is a maximum area? If so, estimate it. () Draw a diagram illustrating te general situation. Introduce notation and lael te diagram wit your symols. (c) Write an expression for te total area. (d) Use te given information to write an equation tat relates te variales. (e) Use part (d) to write te total area as a function of one variale. (f) Finis solving te prolem and compare te answer wit your estimate in part (a). 1. onsider te following prolem: ox wit an open top is to e constructed from a square piece of cardoard, 3 ft wide, y cutting out a square from eac of te four corners and ending up te sides. Find te largest volume tat suc a ox can ave. (a) Draw several diagrams to illustrate te situation, some sort oxes wit large ases and some tall oxes wit small ases. Find te volumes of several suc oxes. Does it appear tat tere is a maximum volume? If so, estimate it. () Draw a diagram illustrating te general situation. Introduce notation and lael te diagram wit your symols. (c) Write an expression for te volume. (d) Use te given information to write an equation tat relates te variales. (e) Use part (d) to write te volume as a function of one variale. (f) Finis solving te prolem and compare te answer wit your estimate in part (a). 13. farmer wants to fence an area of 1.5 million square feet in a rectangular field and ten divide it in alf wit a fence parallel to one of te sides of te rectangle. How can e do tis so as to minimize te cost of te fence? 14. ox wit a square ase and open top must ave a volume of 3 3,000 cm. Find te dimensions of te ox tat mini mize te amount of material used. 15. If 100 cm of material is availale to make a ox wit a square ase and an open top, find te largest possile volume of te ox. 16. rectangular storage container wit an open top is to ave a 3 volume of 10 m. Te lengt of its ase is twice te widt. Material for te ase costs $10 per square meter. Material for te sides costs $6 per square meter. Find te cost of materials for te ceapest suc container. 17. Do Exercise 16 assuming te container as a lid tat is made from te same material as te sides. 18. (a) Sow tat of all te rectangles wit a given area, te one wit smallest perimeter is a square. () Sow tat of all te rectangles wit a given perimeter, te one wit greatest area is a square. 19. Find te point on te line y x 3 tat is closest to te origin. 0. Find te point on te curve y sx tat is closest to te point 3, 0. 1. Find te points on te ellipse 4x y 4 tat are fartest away from te point 1, 0. ;. Find, correct to two decimal places, te coordinates of te point on te curve y sin x tat is closest to te point 4,. 3. Find te dimensions of te rectangle of largest area tat can e inscried in a circle of radius r. 4. Find te area of te largest rectangle tat can e inscried in te ellipse x a y 1. 5. Find te dimensions of te rectangle of largest area tat can e inscried in an equilateral triangle of side L if one side of te rectangle lies on te ase of te triangle. 6. Find te area of te largest trapezoid tat can e inscried in a circle of radius 1 and wose ase is a diameter of te circle. 7. Find te dimensions of te isosceles triangle of largest area tat can e inscried in a circle of radius r. 8. Find te area of te largest rectangle tat can e inscried in a rigt triangle wit legs of lengts 3 cm and 4 cm if two sides of te rectangle lie along te legs. 9. rigt circular cylinder is inscried in a spere of radius r. Find te largest possile volume of suc a cylinder. 30. rigt circular cylinder is inscried in a cone wit eigt and ase radius r. Find te largest possile volume of suc a cylinder. 31. rigt circular cylinder is inscried in a spere of radius r. Find te largest possile surface area of suc a cylinder.
SETION 4.7 OTIMIZTION ROLEMS 333 3. Norman window as te sape of a rectangle surmounted y a semicircle. (Tus te diameter of te semicircle is equal to te widt of te rectangle. See Exercise 6 on page.) If te perimeter of te window is 30 ft, find te dimensions of te window so tat te greatest possile amount of ligt is admitted. 33. Te top and ottom margins of a poster are eac 6 cm and te side margins are eac 4 cm. If te area of printed material on te poster is fixed at 384 cm, find te dimensions of te poster wit te smallest area. 34. poster is to ave an area of 180 in wit 1-inc margins at te ottom and sides and a -inc margin at te top. Wat dimensions will give te largest printed area? 35. piece of wire 10 m long is cut into two pieces. One piece is ent into a square and te oter is ent into an equilateral triangle. How sould te wire e cut so tat te total area enclosed is (a) a maximum? () minimum? 36. nswer Exercise 35 if one piece is ent into a square and te oter into a circle. 37. cylindrical can witout a top is made to contain V cm 3 of liquid. Find te dimensions tat will minimize te cost of te metal to make te can. 38. fence 8 ft tall runs parallel to a tall uilding at a distance of 4 ft from te uilding. Wat is te lengt of te sortest ladder tat will reac from te ground over te fence to te wall of te uilding? 39. cone-saped drinking cup is made from a circular piece of paper of radius R y cutting out a sector and joining te edges and. Find te maximum capacity of suc a cup. 40. cone-saped paper drinking cup is to e made to old 7 cm 3 of water. Find te eigt and radius of te cup tat will use te smallest amount of paper. 41. cone wit eigt is inscried in a larger cone wit eigt H so tat its vertex is at te center of te ase of te larger cone. Sow tat te inner cone as maximum volume wen 1 3 H. 4. n oject wit weigt W is dragged along a orizontal plane y a force acting along a rope attaced to te oject. If te rope makes an angle wit a plane, ten te magnitude of te force is W F sin cos were is a constant called te coefficient of friction. For wat value of is F smallest? R 43. If a resistor of R oms is connected across a attery of E volts wit internal resistance r oms, ten te power ( in watts) in te external resistor is If E and r are fixed ut R varies, wat is te maximum value of te power? 44. For a fis swimming at a speed v relative to te water, te energy expenditure per unit time is proportional to v 3. It is elieved tat migrating fis try to minimize te total energy required to swim a fixed distance. If te fis are swimming against a current u u v, ten te time required to swim a distance L is L v u and te total energy E required to swim te distance is given y E v av 3 were a is te proportionality constant. (a) Determine te value of v tat minimizes E. () Sketc te grap of E. Note: Tis result as een verified experimentally; migrating fis swim against a current at a speed 50% greater tan te current speed. 45. In a eeive, eac cell is a regular exagonal prism, open at one end wit a triedral angle at te oter end as in te figure. It is elieved tat ees form teir cells in suc a way as to minimize te surface area for a given volume, tus using te least amount of wax in cell construction. Examination of tese cells as sown tat te measure of te apex angle is amazingly consistent. ased on te geometry of te cell, it can e sown tat te surface area S is given y S 6s 3 s cot (3s s3 ) csc were s, te lengt of te sides of te exagon, and, te eigt, are constants. (a) alculate ds d. () Wat angle sould te ees prefer? (c) Determine te minimum surface area of te cell ( in terms of s and ). Note: ctual measurements of te angle in eeives ave een made, and te measures of tese angles seldom differ from te calculated value y more tan. rear of cell E R R r s L v u triedral angle front of cell
334 HTER 4 LITIONS OF DIFFERENTITION 46. oat leaves a dock at :00 M and travels due sout at a speed of 0 km. noter oat as een eading due east at 15 km and reaces te same dock at 3:00 M. t wat time were te two oats closest togeter? 47. Solve te prolem in Example 4 if te river is 5 km wide and point is only 5 km downstream from. 48. woman at a point on te sore of a circular lake wit radius mi wants to arrive at te point diametrically opposite on te oter side of te lake in te sortest possile time (see te figure). Se can walk at te rate of 4 mi and row a oat at mi. How sould se proceed? 49. n oil refinery is located on te nort ank of a straigt river tat is km wide. pipeline is to e constructed from te refinery to storage tanks located on te sout ank of te river 6 km east of te refinery. Te cost of laying pipe is $400,000 km over land to a point on te nort ank and $800,000 km under te river to te tanks. To minimize te cost of te pipeline, were sould e located? ; 50. Suppose te refinery in Exercise 49 is located 1 km nort of te river. Were sould e located? 51. Te illumination of an oject y a ligt source is directly proportional to te strengt of te source and inversely proportional to te square of te distance from te source. If two ligt sources, one tree times as strong as te oter, are placed 10 ft apart, were sould an oject e placed on te line etween te sources so as to receive te least illumination? 5. Find an equation of te line troug te point 3, 5 tat cuts off te least area from te first quadrant. 53. Let a and e positive numers. Find te lengt of te sortest line segment tat is cut off y te first quadrant and passes troug te point a,. 54. t wic points on te curve y 1 40x 3 3x 5 does te tangent line ave te largest slope? 55. Wat is te sortest possile lengt of te line segment tat is cut off y te first quadrant and is tangent to te curve y 3 x at some point? 56. Wat is te smallest possile area of te triangle tat is cut off y te first quadrant and wose ypotenuse is tangent to te paraola y 4 x at some point? 57. (a) If x is te cost of producing x units of a commodity, ten te average cost per unit is c x x x. Sow tat if te average cost is a minimum, ten te marginal cost equals te average cost. S () If x 16,000 00x 4x 3, in dollars, find (i) te cost, average cost, and marginal cost at a production level of 1000 units; (ii) te production level tat will minimize te average cost; and (iii) te minimum average cost. 58. (a) Sow tat if te profit x is a maximum, ten te marginal revenue equals te marginal cost. () If x 16,000 500x 1.6x 0.004x 3 is te cost function and p x 1700 7x is te demand function, find te production level tat will maximize profit. 59. aseall team plays in a stadium tat olds 55,000 spectators. Wit ticket prices at $10, te average attendance ad een 7,000. Wen ticket prices were lowered to $8, te average attendance rose to 33,000. (a) Find te demand function, assuming tat it is linear. () How sould ticket prices e set to maximize revenue? 60. During te summer monts Terry makes and sells necklaces on te eac. Last summer e sold te necklaces for $10 eac and is sales averaged 0 per day. Wen e increased te price y $1, e found tat te average decreased y two sales per day. (a) Find te demand function, assuming tat it is linear. () If te material for eac necklace costs Terry $6, wat sould te selling price e to maximize is profit? 61. manufacturer as een selling 1000 flat-screen TVs a week at $450 eac. market survey indicates tat for eac $10 reate offered to te uyer, te numer of TVs sold will increase y 100 per week. (a) Find te demand function. () How large a reate sould te company offer te uyer in order to maximize its revenue? (c) If its weekly cost function is x 68,000 150x, ow sould te manufacturer set te size of te reate in order to maximize its profit? 6. Te manager of a 100-unit apartment complex knows from experience tat all units will e occupied if te rent is $800 per mont. market survey suggests tat, on average, one additional unit will remain vacant for eac $10 increase in rent. Wat rent sould te manager carge to maximize revenue? 63. Sow tat of all te isosceles triangles wit a given perimeter, te one wit te greatest area is equilateral. 64. Te frame for a kite is to e made from six pieces of wood. Te four exterior pieces ave een cut wit te lengts indicated in te figure. To maximize te area of te kite, ow long sould te diagonal pieces e? a a
SETION 4.7 OTIMIZTION ROLEMS 335 ; 65. point needs to e located somewere on te line D so tat te total lengt L of cales linking to te points,, and is minimized (see te figure). Express L as a function of x and use te graps of L and dl dx to estimate te minimum value of L. etween te poles and ten to te top of te second pole as in te figure. Sow tat te sortest lengt of suc a rope occurs wen 1. S 5 m Q R T 66. Te grap sows te fuel consumption c of a car (measured in gallons per our) as a function of te speed v of te car. t very low speeds te engine runs inefficiently, so initially c decreases as te speed increases. ut at ig speeds te fuel consumption increases. You can see tat c v is minimized for tis car wen v 30 mi. However, for fuel efficiency, wat must e minimized is not te consumption in gallons per our ut rater te fuel consumption in gallons per mile. Let s call tis consumption G. Using te grap, estimate te speed at wic G as its minimum value. c m 3 m D 69. Te upper rigt-and corner of a piece of paper, 1 in. y 8 in., as in te figure, is folded over to te ottom edge. How would you fold it so as to minimize te lengt of te fold? In oter words, ow would you coose x to minimize y? 8 1 70. steel pipe is eing carried down a allway 9 ft wide. t te end of te all tere is a rigt-angled turn into a narrower allway 6 ft wide. Wat is te lengt of te longest pipe tat can e carried orizontally around te corner? y x 67. Let v 1 e te velocity of ligt in air and v te velocity of ligt in water. ccording to Fermat s rinciple, a ray of ligt will travel from a point in te air to a point in te water y a pat tat minimizes te time taken. Sow tat were 1 (te angle of incidence) and (te angle of refraction) are as sown. Tis equation is known as Snell s Law. 0 0 40 60 sin 1 sin v 1 v 6 9 71. n oserver stands at a point, one unit away from a track. Two runners start at te point S in te figure and run along te track. One runner runs tree times as fast as te oter. Find te maximum value of te oserver s angle of sigt etween te runners. [Hint: Maximize tan.] 1 68. Two vertical poles Q and ST are secured y a rope RS going from te top of te first pole to a point R on te ground S
336 HTER 4 LITIONS OF DIFFERENTITION 7. rain gutter is to e constructed from a metal seet of widt 30 cm y ending up one-tird of te seet on eac side troug an angle. How sould e cosen so tat te gutter will carry te maximum amount of water? sel wit radius r 1 rancing at an angle into a smaller vessel wit radius r. 10 cm 10 cm 10 cm 73. Were sould te point e cosen on te line segment so as to maximize te angle? vascular rancing r a r 3 (a) Use oiseuille s Law to sow tat te total resistance of te lood along te pat is a cot csc R 4 4 r 1 r 5 were a and are te distances sown in te figure. () rove tat tis resistance is minimized wen 74. painting in an art gallery as eigt and is ung so tat its lower edge is a distance d aove te eye of an oserver (as in te figure). How far from te wall sould te oserver stand to get te est view? (In oter words, were sould te oserver stand so as to maximize te angle sutended at is eye y te painting?) cos r 4 r 4 1 (c) Find te optimal rancing angle (correct to te nearest degree) wen te radius of te smaller lood vessel is twotirds te radius of te larger vessel. 75. Find te maximum area of a rectangle tat can e circum - scried aout a given rectangle wit lengt L and widt W. [Hint: Express te area as a function of an angle.] 76. Te lood vascular system consists of lood vessels (arteries, arterioles, capillaries, and veins) tat convey lood from te eart to te organs and ack to te eart. Tis system sould work so as to minimize te energy expended y te eart in pumping te lood. In particular, tis energy is reduced wen te resistance of te lood is lowered. One of oiseuille s Laws gives te resistance R of te lood as R L r 4 were L is te lengt of te lood vessel, r is te radius, and is a positive constant determined y te viscosity of te lood. (oiseuille estalised tis law experimentally, ut it also follows from Equation 8.4..) Te figure sows a main lood ves- d Manfred Kage / eter rnold Images / otolirary 77. Ornitologists ave determined tat some species of irds tend to avoid fligts over large odies of water during dayligt ours. It is elieved tat more energy is required to fly over water tan over land ecause air generally rises over land and falls over water during te day. ird wit tese tendencies is released from an island tat is 5 km from te nearest point on a straigt soreline, flies to a point on te soreline, and ten flies along te soreline to its nesting area D. ssume tat te ird instinctively cooses a pat tat will minimize its energy expenditure. oints and D are 13 km apart. (a) In general, if it takes 1.4 times as muc energy to fly over water as it does over land, to wat point sould te ird
LIED ROJET THE SHE OF N 337 fly in order to minimize te total energy expended in returning to its nesting area? () Let W and L denote te energy (in joules) per kilometer flown over water and land, respectively. Wat would a large value of te ratio W L mean in terms of te ird s fligt? Wat would a small value mean? Determine te ratio W L corresponding to te minimum expenditure of energy. (c) Wat sould te value of W L e in order for te ird to fly directly to its nesting area D? Wat sould te value of W L e for te ird to fly to and ten along te sore to D? (d) If te ornitologists oserve tat irds of a certain species reac te sore at a point 4 km from, ow many times more energy does it take a ird to fly over water tan over land? island ; 78. Two ligt sources of identical strengt are placed 10 m apart. n oject is to e placed at a point on a line parallel to te line joining te ligt sources and at a distance d meters from it (see te figure). We want to locate on so tat te intensity of illumination is minimized. We need to use te fact tat te intensity of illumination for a single source is directly proportional to te strengt of te source and inversely proportional to te square of te distance from te source. (a) Find an expression for te intensity I x at te point. () If d 5 m, use graps of I x and I x to sow tat te intensity is minimized wen x 5 m, tat is, wen is at te midpoint of. (c) If d 10 m, sow tat te intensity (peraps surpris - ingly) is not minimized at te midpoint. (d) Somewere etween d 5 m and d 10 m tere is a transitional value of d at wic te point of minimal illumination aruptly canges. Estimate tis value of d y grapical metods. Ten find te exact value of d. 5 km 13 km D nest x 10 m d LIED ROJET THE SHE OF N r In tis project we investigate te most economical sape for a can. We first interpret tis to mean tat te volume V of a cylindrical can is given and we need to find te eigt and radius r tat minimize te cost of te metal to make te can (see te figure). If we disregard any waste metal in te manufacturing process, ten te prolem is to minimize te surface area of te cylinder. We solved tis prolem in Example in Section 4.7 and we found tat r; tat is, te eigt sould e te same as te diameter. ut if you go to your cupoard or your supermarket wit a ruler, you will discover tat te eigt is usually greater tan te diameter and te ratio r varies from up to aout 3.8. Let s see if we can explain tis penomenon. 1. Te material for te cans is cut from seets of metal. Te cylindrical sides are formed y ending rectangles; tese rectangles are cut from te seet wit little or no waste. ut if te top and ottom discs are cut from squares of side r (as in te figure), tis leaves considerale waste metal, wic may e recycled ut as little or no value to te can makers. If tis is te case, sow tat te amount of metal used is minimized wen Discs cut from squares r 8.55 ; Graping calculator or computer required