336 HTER 4 pplications of Differentiation Now let s consider marketing. Let psxd e the price per unit that the company can charge if it sells x units. Then p is called the demand function (or price function) and we would expect it to e a decreasing function of x. (More units sold corresponds to a lower price.) If x units are sold and the price per unit is psxd, then the total revenue is Rsxd quantity 3 price xpsxd and R is called the revenue function. The derivative R9 of the revenue function is called the marginal revenue function and is the rate of change of revenue with respect to the num er of units sold. If x units are sold, then the total profit is sxd Rsxd sxd and is called the profit function. The marginal profit function is 9, the derivative of the profit function. In Exercises 59 63 you are asked to use the marginal cost, revenue, and profit functions to minimize costs and maximize revenues and profits. EXMLE 6 store has een selling 00 flat-screen TVs a week at $350 each. market survey indicates that for each $10 reate offered to uyers, the numer of TVs sold will increase y 0 a week. Find the demand function and the revenue function. How large a reate should the store offer to maximize its revenue? SOLUTION If x is the numer of TVs sold per week, then the weekly increase in sales is x 00. For each increase of 0 units sold, the price is decreased y $10. So for each additional unit sold, the decrease in price will e 1 0 3 10 and the demand function is The revenue function is psxd 350 10 0 sx 00d 450 1 x Rsxd xpsxd 450x 1 x Since R9sxd 450 x, we see that R9sxd 0 when x 450. This value of x gives an asolute maximum y the First Derivative Test (or simply y oserving that the graph of R is a paraola that opens downward). The corresponding price is ps450d 450 1 s450d 5 and the reate is 350 5 15. Therefore, to maximize revenue, the store should offer a reate of $15. 1. onsider the following prolem: Find two numers whose sum is 3 and whose product is a maximum. (a) Make a tale of values, like the one at the right, so that the sum of the numers in the first two columns is always 3. On the asis of the evidence in your tale, estimate the answer to the prolem. () Use calculus to solve the prolem and compare with your answer to part (a). First numer Second numer roduct 1 1 4 3 0 60...
SETION 4.7 Optimization rolems 337. Find two numers whose difference is 100 and whose product is a minimum. 3. Find two positive numers whose product is 100 and whose sum is a minimum. 4. The sum of two positive numers is 16. What is the smallest possile value of the sum of their squares? 5. What is the maximum vertical distance etween the line y x 1 and the paraola y x for 1 < x <? 6. What is the minimum vertical distance etween the paraolas y x 1 1 and y x x? 7. Find the dimensions of a rectangle with perimeter 100 m whose area is as large as possile. 8. Find the dimensions of a rectangle with area 1000 m whose perimeter is as small as possile. 9. model used for the yield Y of an agricultural crop as a function of the nitrogen level N in the soil (measured in appropriate units) is Y kn 1 1 N where k is a positive constant. What nitrogen level gives the est yield? 10. The rate sin mg caronym 3 yhd at which photosynthesis takes place for a species of phytoplankton is modeled y the function 100I I 1 I 1 4 where I is the light intensity (measured in thousands of footcandles). For what light intensity is a maximum? 11. onsider the following prolem: farmer with 750 ft of fencing wants to enclose a rectangular area and then divide it into four pens with fencing parallel to one side of the rectangle. What is the largest possile total area of the four pens? (a) Draw several diagrams illustrating the situation, some with shallow, wide pens and some with deep, narrow pens. Find the total areas of these configurations. Does it appear that there is a maximum area? If so, estimate it. () Draw a diagram illustrating the general situation. Introduce notation and lael the diagram with your symols. (c) Write an expression for the total area. (d) Use the given information to write an equation that relates the variales. (e) Use part (d) to write the total area as a function of one variale. (f ) Finish solving the prolem and compare the answer with your estimate in part (a). 1. onsider the following prolem: ox with an open top is to e constructed from a square piece of cardoard, 3 ft wide, y cutting out a square from each of the four corners and end- ing up the sides. Find the largest volume that such a ox can have. (a) Draw several diagrams to illustrate the situation, some short oxes with large ases and some tall oxes with small ases. Find the volumes of several such oxes. Does it appear that there is a maximum volume? If so, estimate it. () Draw a diagram illustrating the general situation. Introduce notation and lael the diagram with your symols. (c) Write an expression for the volume. (d) Use the given information to write an equation that relates the variales. (e) Use part (d) to write the volume as a function of one variale. (f ) Finish solving the prolem and compare the answer with your estimate in part (a). 13. farmer wants to fence in an area of 1.5 million square feet in a rectangular field and then divide it in half with a fence parallel to one of the sides of the rectangle. How can he do this so as to minimize the cost of the fence? 14. ox with a square ase and open top must have a volume of 3,000 cm 3. Find the dimensions of the ox that minimize the amount of material used. 15. If 100 cm of material is availale to make a ox with a square ase and an open top, find the largest possile volume of the ox. 16. rectangular storage container with an open top is to have a volume of 10 m 3. The length of its ase is twice the width. Material for the ase costs $10 per square meter. Material for the sides costs $6 per square meter. Find the cost of materials for the cheapest such container. 17. Do Exercise 16 assuming the container has a lid that is made from the same material as the sides. 18. farmer wants to fence in a rectangular plot of land adjacent to the north wall of his arn. No fencing is needed along the arn, and the fencing along the west side of the plot is shared with a neighor who will split the cost of that portion of the fence. If the fencing costs $0 per linear foot to install and the farmer is not willing to spend more than $5000, find the dimensions for the plot that would enclose the most area. 19. If the farmer in Exercise 18 wants to enclose 8000 square feet of land, what dimensions will minimize the cost of the fence? 0. (a) Show that of all the rectangles with a given area, the one with smallest perimeter is a square. () Show that of all the rectangles with a given perimeter, the one with greatest area is a square. 1. Find the point on the line y x 1 3 that is closest to the origin.. Find the point on the curve y sx that is closest to the point s3, 0d.
338 HTER 4 pplications of Differentiation 3. Find the points on the ellipse 4x 1 y 4 that are farthest away from the point s1, 0d. 4. Find, correct to two decimal places, the coordinates of the point on the curve y sin x that is closest to the point s4, d. 5. Find the dimensions of the rectangle of largest area that can e inscried in a circle of radius r. 6. Find the area of the largest rectangle that can e inscried in the ellipse x ya 1 y y 1. 7. Find the dimensions of the rectangle of largest area that can e inscried in an equilateral triangle of side L if one side of the rectangle lies on the ase of the triangle. 8. Find the area of the largest trapezoid that can e inscried in a circle of radius 1 and whose ase is a diameter of the circle. 9. Find the dimensions of the isosceles triangle of largest area that can e inscried in a circle of radius r. 30. If the two equal sides of an isosceles triangle have length a, find the length of the third side that maximizes the area of the triangle. 31. right circular cylinder is inscried in a sphere of radius r. Find the largest possile volume of such a cylinder. 3. right circular cylinder is inscried in a cone with height h and ase radius r. Find the largest possile volume of such a cylinder. 33. right circular cylinder is inscried in a sphere of radius r. Find the largest possile surface area of such a cylinder. 34. Norman window has the shape of a rectangle surmounted y a semicircle. (Thus the diameter of the semicircle is equal to the width of the rectangle. See Exercise 1.1.6.) If the perimeter of the window is 30 ft, find the dimensions of the window so that the greatest possile amount of light is admitted. 35. The top and ottom margins of a poster are each 6 cm and the side margins are each 4 cm. If the area of printed material on the poster is fixed at 384 cm, find the dimensions of the poster with the smallest area. 36. poster is to have an area of 180 in with 1-inch margins at the ottom and sides and a -inch margin at the top. What dimensions will give the largest printed area? 37. piece of wire 10 m long is cut into two pieces. One piece is ent into a square and the other is ent into an equilateral triangle. How should the wire e cut so that the total area enclosed is (a) a maximum? () minimum? 38. nswer Exercise 37 if one piece is ent into a square and the other into a circle. 39. If you are offered one slice from a round pizza (in other words, a sector of a circle) and the slice must have a perimeter of 3 inches, what diameter pizza will reward you with the largest slice? 40. fence 8 ft tall runs parallel to a tall uilding at a distance of 4 ft from the uilding. What is the length of the shortest ladder that will reach from the ground over the fence to the wall of the uilding? 41. cone-shaped drinking cup is made from a circular piece of paper of radius R y cutting out a sector and joining the edges and. Find the maximum capacity of such a cup. 4. cone-shaped paper drinking cup is to e made to hold 7 cm 3 of water. Find the height and radius of the cup that will use the smallest amount of paper. 43. cone with height h is inscried in a larger cone with height H so that its vertex is at the center of the ase of the larger cone. Show that the inner cone has maximum volume when h 1 3 H. 44. n oject with weight W is dragged along a horizontal plane y a force acting along a rope attached to the oject. If the rope makes an angle with a plane, then the magnitude of the force is F R W sin 1 cos where is a constant called the coefficient of friction. For what value of is F smallest? 45. If a resistor of R ohms is connected across a attery of E volts with internal resistance r ohms, then the power (in watts) in the external resistor is E R sr 1 rd If E and r are fixed ut R varies, what is the maximum value of the power? 46. For a fish swimming at a speed v relative to the water, the energy expenditure per unit time is proportional to v 3. It is elieved that migrating fish try to minimize the total energy required to swim a fixed distance. If the fish are swimming against a current u su, vd, then the time
SETION 4.7 Optimization rolems 339 required to swim a distance L is Lysv ud and the total energy E required to swim the distance is given y Esvd av 3? L v u where a is the proportionality constant. (a) Determine the value of v that minimizes E. () Sketch the graph of E. Note: This result has een verified experimentally migrating fish swim against a current at a speed 50% greater than the current speed. 47. In a eehive, each cell is a regular hexagonal prism, open at one end with a trihedral angle at the other end as in the figure. It is elieved that ees form their cells in such a way as to min imize the surface area for a given side length and height, thus using the least amount of wax in cell construction. Examination of these cells has shown that the measure of the apex angle is amazingly consistent. ased on the geometry of the cell, it can e shown that the surface area S is given y S 6sh 3 s cot 1 (3s s3y) csc where s, the length of the sides of the hexagon, and h, the height, are constants. (a) alculate dsyd. () What angle should the ees prefer? (c) Determine the minimum surface area of the cell (in terms of s and h). Note: ctual measurements of the angle in eehives have een made, and the measures of these angles seldom differ from the calculated value y more than 8. rear of cell s h trihedral angle front of cell 48. oat leaves a dock at :00 pm and travels due south at a speed of 0 kmyh. nother oat has een heading due east at 15 kmyh and reaches the same dock at 3:00 pm. t what time were the two oats closest together? 49. Solve the prolem in Example 4 if the river is 5 km wide and point is only 5 km downstream from. 50. woman at a point on the shore of a circular lake with radius mi wants to arrive at the point diametrically opposite on the other side of the lake in the shortest possile time (see the figure). She can walk at the rate of 4 miyh and row a oat at miyh. How should she proceed? 51. n oil refinery is located on the north ank of a straight river that is km wide. pipeline is to e constructed from the refinery to storage tanks located on the south ank of the river 6 km east of the refinery. The cost of laying pipe is $400,000ykm over land to a point on the north ank and $800,000ykm under the river to the tanks. To minimize the cost of the pipeline, where should e located? 5. Suppose the refinery in Exercise 51 is located 1 km north of the river. Where should e located? 53. The illumination of an oject y a light source is directly proportional to the strength of the source and inversely proportional to the square of the distance from the source. If two light sources, one three times as strong as the other, are placed 10 ft apart, where should an oject e placed on the line etween the sources so as to receive the least illumination? 54. Find an equation of the line through the point s3, 5d that cuts off the least area from the first quadrant. 55. Let a and e positive numers. Find the length of the shortest line segment that is cut off y the first quadrant and passes through the point sa, d. 56. t which points on the curve y 1 1 40x 3 3x 5 does the tangent line have the largest slope? 57. What is the shortest possile length of the line segment that is cut off y the first quadrant and is tangent to the curve y 3yx at some point? 58. What is the smallest possile area of the triangle that is cut off y the first quadrant and whose hypotenuse is tangent to the paraola y 4 x at some point? 59. (a) If sxd is the cost of producing x units of a commodity, then the average cost per unit is csxd sxdyx. Show that if the average cost is a minimum, then the marginal cost equals the average cost. () If sxd 16,000 1 00x 1 4x 3y, in dollars, find (i) the cost, average cost, and marginal cost at a production level of 1000 units (ii) the production level that will minimize the average cost and (iii) the minimum average cost.
340 HTER 4 pplications of Differentiation 60. (a) Show that if the profit sxd is a maximum, then the marginal revenue equals the marginal cost. () If sxd 16,000 1 500x 1.6x 1 0.004x 3 is the cost function and psxd 1700 7x is the demand function, find the production level that will maximize profit. 61. aseall team plays in a stadium that holds 55,000 spectators. With ticket prices at $10, the average attendance had een 7,000. When ticket prices were lowered to $8, the average attendance rose to 33,000. (a) Find the demand function, assuming that it is linear. () How should ticket prices e set to maximize revenue? 6. During the summer months Terry makes and sells necklaces on the each. Last summer he sold the necklaces for $10 each and his sales averaged 0 per day. When he increased the price y $1, he found that the average decreased y two sales per day. (a) Find the demand function, assuming that it is linear. () If the material for each necklace costs Terry $6, what should the selling price e to maximize his profit? 63. retailer has een selling 100 talet computers a week at $350 each. The marketing department estimates that an additional 80 talets will sell each week for every $10 that the price is lowered. (a) Find the demand function. () What should the price e set at in order to maximize revenue? (c) If the retailer s weekly cost function is sxd 35,000 1 10x what price should it choose in order to maximize its profit? 64. company operates 16 oil wells in a designated area. Each pump, on average, extracts 40 arrels of oil daily. The company can add more wells ut every added well reduces the average daily ouput of each of the wells y 8 arrels. How many wells should the company add in order to maximize daily production? 65. Show that of all the isosceles triangles with a given perimeter, the one with the greatest area is equilateral. 66. onsider the situation in Exercise 51 if the cost of laying pipe under the river is consideraly higher than the cost of laying pipe over land ($400,000ykm). You may suspect that in some instances, the minimum distance possile under the river should e used, and should e located 6 km from the refinery, directly across from the storage tanks. Show that this is never the case, no matter what the under river cost is. 67. onsider the tangent line to the ellipse x a 1 y 1 at a point s p, qd in the first quadrant. (a) Show that the tangent line has x-intercept a yp and y-intercept yq. () Show that the portion of the tangent line cut off y the coordinate axes has minimum length a 1. S (c) Show that the triangle formed y the tangent line and the coordinate axes has minimum area a. 68. The frame for a kite is to e made from six pieces of wood. The four exterior pieces have een cut with the lengths indicated in the figure. To maximize the area of the kite, how long should the diagonal pieces e? a a 69. point needs to e located somewhere on the line D so that the total length L of cales linking to the points,. and is minimized (see the figure). Express L as a function of x and use the graphs of L and dlydx to estimate the minimum value of L. m 3 m D 5 m 70. The graph shows the fuel consumption c of a car (measured in gallons per hour) as a function of the speed v of the car. t very low speeds the engine runs inefficiently, so initially c decreases as the speed increases. ut at high speeds the fuel consumption increases. You can see that csvd is minimized for this car when v < 30 miyh. However, for fuel efficiency, what must e minimized is not the consumption in gallons per hour ut rather the fuel consumption in gallons per mile. Let s call this consumption G. Using the graph, estimate the speed at which G has its minimum value. c 0 0 40 60 71. Let v 1 e the velocity of light in air and v the velocity of light in water. ccording to Fermat s rinciple, a ray of light will travel from a point in the air to a point in the water y a path that minimizes the time taken. Show that sin 1 sin v1 v
SETION 4.7 Optimization rolems 341 where 1 (the angle of incidence) and (the angle of refraction) are as shown. This equation is known as Snell s Law. 75. n oserver stands at a point, one unit away from a track. Two runners start at the point S in the figure and run along the track. One runner runs three times as fast as the other. Find the maximum value of the oserver s angle of sight etween the runners. 1 7. Two vertical poles Q and ST are secured y a rope RS going from the top of the first pole to a point R on the ground etween the poles and then to the top of the second pole as in the figure. Show that the shortest length of such a rope occurs when 1. S S 76. rain gutter is to e constructed from a metal sheet of width 30 cm y ending up one-third of the sheet on each side through an angle. How should e chosen so that the gutter will carry the maximum amount of water? Q R T 73. The upper right-hand corner of a piece of paper, 1 in. y 8 in., as in the figure, is folded over to the ottom edge. How would you fold it so as to minimize the length of the fold? In other words, how would you choose x to minimize y? 1 10 cm 10 cm 10 cm 77. Where should the point e chosen on the line segment so as to maximize the angle? 3 8 y x 5 74. steel pipe is eing carried down a hallway 9 ft wide. t the end of the hall there is a right-angled turn into a narrower hallway 6 ft wide. What is the length of the longest pipe that can e carried horizontally around the corner? 78. painting in an art gallery has height h and is hung so that its lower edge is a distance d aove the eye of an oserver (as in the figure). How far from the wall should the oserver stand to get the est view? (In other words, where should the oserver stand so as to maximize the angle sutended at his eye y the painting?) h 6 d 9 79. Find the maximum area of a rectangle that can e circumscried aout a given rectangle with length L and width W. [Hint: Express the area as a function of an angle.]
34 HTER 4 pplications of Differentiation 80. The lood vascular system consists of lood vessels (arteries, arterioles, capillaries, and veins) that convey lood from the heart to the organs and ack to the heart. This system should work so as to minimize the energy expended y the heart in pumping the lood. In particular, this energy is reduced when the resistance of the lood is lowered. One of oiseuille s Laws gives the resistance R of the lood as R L r 4 where L is the length of the lood vessel, r is the radius, and is a positive constant determined y the viscosity of the lood. (oiseuille estalished this law experimentally, ut it also follows from Equation 8.4..) The figure shows a main lood vessel with radius r 1 ranching at an angle into a smaller vessel with radius r. fly in order to minimize the total energy expended in returning to its nesting area? () Let W and L denote the energy (in joules) per kilometer flown over water and land, respectively. What would a large value of the ratio WyL mean in terms of the ird s flight? What would a small value mean? Determine the ratio WyL corresponding to the minimum expenditure of energy. (c) What should the value of WyL e in order for the ird to fly directly to its nesting area D? What should the value of WyL e for the ird to fly to and then along the shore to D? (d) If the ornithologists oserve that irds of a certain species reach the shore at a point 4 km from, how many times more energy does it take a ird to fly over water than over land? island vascular ranching r r 5 km 13 km D nest a (a) Use oiseuille s Law to show that the total resistance of the lood along the path is a cot R S 4 1 r 1 csc 4 r D where a and are the distances shown in the figure. () rove that this resistance is minimized when cos r 4 r 4 1 (c) Find the optimal ranching angle (correct to the nearest degree) when the radius of the smaller lood vessel is two-thirds the radius of the larger vessel. 81. Ornithologists have determined that some species of irds tend to avoid flights over large odies of water during daylight hours. It is elieved that more energy is required to fly over water than over land ecause air generally rises over land and falls over water during the day. ird with these tendencies is released from an island that is 5 km from the nearest point on a straight shoreline, flies to a point on the shoreline, and then flies along the shoreline to its nesting area D. ssume that the ird instinctively chooses a path that will minimize its energy expenditure. oints and D are 13 km apart. (a) In general, if it takes 1.4 times as much energy to fly over water as it does over land, to what point should the ird 8. Two light sources of identical strength are placed 10 m apart. n oject is to e placed at a point on a line,, paral lel to the line joining the light sources and at a distance d meters from it (see the figure). We want to locate on, so that the intensity of illumination is minimized. We need to use the fact that the intensity of illumination for a single source is directly proportional to the strength of the source and inversely proportional to the square of the distance from the source. (a) Find an expression for the intensity Isxd at the point. () If d 5 m, use graphs of Isxd and I9sxd to show that the intensity is minimized when x 5 m, that is, when is at the midpoint of,. (c) If d 10 m, show that the intensity (perhaps surprisingly) is not minimized at the midpoint. (d) Somewhere etween d 5 m and d 10 m there is a transitional value of d at which the point of minimal illumination aruptly changes. Estimate this value of d y graphical methods. Then find the exact value of d. x 10 m d