If the farmer in Exercise 18 wants to enclose 8000 square feet of land, what dimensions will minimize the cost of the fence? Exercise 18: A farmer wants to fence in a rectangular plot of land adjacent to the north wall of his barn. No fencing is needed along the barn, and the fencing along the west side of the plot is shared with a neighbor who will split the cost of that portion of the fence. If the fencing costs $20 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.
> An observer stands at a point P, 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 observer’s angle of sig
> A steel pipe is being carried down a hallway 9 ft wide. At 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 be carried horizontally around the corner?
> The upper right-hand corner of a piece of paper, 12 in. by 8 in., as in the figure, is folded over to the bottom 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? 12 - y 8
> Two vertical poles PQ and ST are secured by a rope PRS going from the top of the first pole to a point R on the ground between 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 &Ic
> 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. At very low speeds the engine runs inefficiently, so initially c decreases as the speed increases. But at high speeds the fuel consump
> A point P needs to be located somewhere on the line AD so that the total length L of cables linking P to the points A, B. and C is minimized (see the figure). Express L as a function of x = |AP | and use the graphs of L and dL/dx to estimate the minimum
> The frame for a kite is to be made from six pieces of wood. The four exterior pieces have been cut with the lengths indicated in the figure. To maximize the area of the kite, how long should the diagonal pieces be? b a a b
> Consider the situation in Exercise 51 if the cost of laying pipe under the river is considerably higher than the cost of laying pipe over land ($400,000ykm). You may suspect that in some instances, the minimum distance possible under the river should be
> A retailer has been selling 1200 tablet computers a week at $350 each. The marketing department estimates that an additional 80 tablets will sell each week for every $10 that the price is lowered. (a) Find the demand function. (b) What should the price b
> During the summer months Terry makes and sells necklaces on the beach. Last summer he sold the necklaces for $10 each and his sales averaged 20 per day. When he increased the price by $1, he found that the average decreased by two sales per day. (a) Find
> A baseball team plays in a stadium that holds 55,000 spectators. With ticket prices at $10, the average attendance had been 27,000. When ticket prices were lowered to $8, the average attendance rose to 33,000. (a) Find the demand function, assuming that
> (a) Show that if the profit P(x) is a maximum, then the marginal revenue equals the marginal cost. (b) If C(x) = 16,000 + 500x - 1.6x2 + 0.004x3 is the cost function and p(x) = 1700 - 7x is the demand function, find the production level that will maximiz
> (a) If C(x) is the cost of producing x units of a commodity, then the average cost per unit is c(x) = C(x)/x. Show that if the average cost is a minimum, then the marginal cost equals the average cost. (b) If C(x) = 16,000 + 200x + 4x3/2, in dollars, fin
> What is the smallest possible area of the triangle that is cut off by the first quadrant and whose hypotenuse is tangent to the parabola y = 4 - x2 at some point?
> What is the shortest possible length of the line segment that is cut off by the first quadrant and is tangent to the curve y = 3/x at some point?
> At which points on the curve y = 1 + 40x3 - 3x5 does the tangent line have the largest slope?
> Let a and b be positive numbers. Find the length of the shortest line segment that is cut off by the first quadrant and passes through the point (a, b).
> Find an equation of the line through the point (3, 5) that cuts off the least area from the first quadrant.
> The illumination of an object by 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 f
> Suppose the refinery in Exercise 51 is located 1 km north of the river. Where should P be located? Exercise 51: An oil refinery is located on the north bank of a straight river that is 2 km wide. A pipeline is to be constructed from the refinery to stor
> An oil refinery is located on the north bank of a straight river that is 2 km wide. A pipeline is to be constructed from the refinery to storage tanks located on the south bank of the river 6 km east of the refinery. The cost of laying pipe is $400,000/k
> A woman at a point A on the shore of a circular lake with radius 2 mi wants to arrive at the point C diametrically opposite A on the other side of the lake in the shortest possible time (see the figure). She can walk at the rate of 4 mi/h and row a boat
> Solve the problem in Example 4 if the river is 5 km wide and point B is only 5 km downstream from A. Example 4: A man launches his boat from point A on a bank of a straight river, 3 km wide, and wants to reach point B, 8 km downstream on the opposite ba
> A boat leaves a dock at 2:00 pm and travels due south at a speed of 20 km/h. Another boat has been heading due east at 15 km/h and reaches the same dock at 3:00 pm. At what time were the two boats closest together?
> Differentiate the function. F(t) = (ln t)2 sin t
> A cone with height h is inscribed in a larger cone with height H so that its vertex is at the center of the base of the larger cone. Show that the inner cone has maximum volume when h = 1/3 H.
> A cone-shaped paper drinking cup is to be made to hold 27 cm3 of water. Find the height and radius of the cup that will use the smallest amount of paper.
> A cone-shaped drinking cup is made from a circular piece of paper of radius R by cutting out a sector and joining the edges CA and CB. Find the maximum capacity of such a cup. A B R C
> A fence 8 ft tall runs parallel to a tall building at a distance of 4 ft from the building. What is the length of the shortest ladder that will reach from the ground over the fence to the wall of the building?
> 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 32 inches, what diameter pizza will reward you with the largest slice?
> Answer Exercise 37 if one piece is bent into a square and the other into a circle. Exercise 37: A piece of wire 10 m long is cut into two pieces. One piece is bent into a square and the other is bent into an equilateral triangle. How should the wire be
> A piece of wire 10 m long is cut into two pieces. One piece is bent into a square and the other is bent into an equilateral triangle. How should the wire be cut so that the total area enclosed is (a) a maximum? (b) A minimum?
> A poster is to have an area of 180 in2 with 1-inch margins at the bottom and sides and a 2-inch margin at the top. What dimensions will give the largest printed area?
> The top and bottom 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 cm2, find the dimensions of the poster with the smallest area.
> A Norman window has the shape of a rectangle surmounted by a semicircle. (Thus the diameter of the semicircle is equal to the width of the rectangle. See Exercise 1.1.62.) If the perimeter of the window is 30 ft, find the dimensions of the window so that
> A right circular cylinder is inscribed in a sphere of radius r. Find the largest possible surface area of such a cylinder.
> A right circular cylinder is inscribed in a cone with height h and base radius r. Find the largest possible volume of such a cylinder.
> A right circular cylinder is inscribed in a sphere of radius r. Find the largest possible volume of such a cylinder.
> 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.
> Find the dimensions of the isosceles triangle of largest area that can be inscribed in a circle of radius r.
> Find the area of the largest trapezoid that can be inscribed in a circle of radius 1 and whose base is a diameter of the circle.
> Find the dimensions of the rectangle of largest area that can be inscribed in an equilateral triangle of side L if one side of the rectangle lies on the base of the triangle.
> Differentiate the function. g(x) = ln(xe-2x)
> Find the area of the largest rectangle that can be inscribed in the ellipse x2/a2 + y2/b2 = 1.
> Find the dimensions of the rectangle of largest area that can be inscribed in a circle of radius r.
> Find, correct to two decimal places, the coordinates of the point on the curve y = sin x that is closest to the point (4, 2).
> Find the points on the ellipse 4x2 + y2 = 4 that are farthest away from the point (1, 0).
> Find the point on the line y = 2x + 3 that is closest to the origin.
> (a) Show that of all the rectangles with a given area, the one with smallest perimeter is a square. (b) Show that of all the rectangles with a given perimeter, the one with greatest area is a square.
> A farmer wants to fence in a rectangular plot of land adjacent to the north wall of his barn. No fencing is needed along the barn, and the fencing along the west side of the plot is shared with a neighbor who will split the cost of that portion of the fe
> Do Exercise 16 assuming the container has a lid that is made from the same material as the sides. Exercise 16: A rectangular storage container with an open top is to have a volume of 10 m3. The length of its base is twice the width. Material for the bas
> A rectangular storage container with an open top is to have a volume of 10 m3. The length of its base is twice the width. Material for the base costs $10 per square meter. Material for the sides costs $6 per square meter. Find the cost of materials for t
> If 1200 cm2 of material is available to make a box with a square base and an open top, find the largest possible volume of the box.
> A box with a square base and open top must have a volume of 32,000 cm3. Find the dimensions of the box that minimize the amount of material used.
> A 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?
> Consider the following problem: A box with an open top is to be constructed from a square piece of cardboard, 3 ft wide, by cutting out a square from each of the four corners and bending up the sides. Find the largest volume that such a box can have. (a)
> Consider the following problem: A 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 possible total area of the four pens? (a) Draw s
> Find the dimensions of a rectangle with area 1000 m2 whose perimeter is as small as possible.
> Find the dimensions of a rectangle with perimeter 100 m whose area is as large as possible.
> Find expressions for the first five derivatives of f (x) = x2ex. Do you see a pattern in these expressions? Guess a formula for f(n) (x) and prove it using mathematical induction.
> What is the minimum vertical distance between the parabolas y = x2 + 1 and y = x - x2?
> Produce graphs of f that reveal all the important aspects of the curve. In particular, you should use graphs of f ’ and f ’’ to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points. f(x) = 6 sin x
> Produce graphs of f that reveal all the important aspects of the curve. In particular, you should use graphs of f ’ and f ’’ to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points. f (x) = x6 - 5
> Produce graphs of f that reveal all the important aspects of the curve. In particular, you should use graphs of f ’ and f ’’ to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points. f (x) = 22x6 +
> Produce graphs of f that reveal all the important aspects of the curve. In particular, you should use graphs of f ’ and f ’’ to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points. f (x) = x5 - 5
> (a) Investigate the family of polynomials given by the equation f (x) = 2x3 + cx2 - 2x. For what values of c does the curve have maximum and minimum points? (b) Show that the minimum and maximum points of every curve in the family lie on the curve y = x
> (a) Investigate the family of polynomials given by the equation f (x) = cx4 - 2x2 + 1. For what values of c does the curve have minimum points? (b) Show that the minimum and maximum points of every curve in the family lie on the parabola y = 1 - x2. Illu
> Investigate the family of curves given by the equation f (x) = x4 + cx2 + x. Start by determining the transitional value of c at which the number of inflection points changes. Then graph several members of the family to see what shapes are possible. Ther
> Differentiate the function. f(x) = log 10 (1 + cos x)
> Investigate the family of curves given by f (x) = xe-cx, where c is a real number. Start by computing the limits as x ( ±∞. Identify any transitional values of c where the basic shape changes. What happens to the maximum or minimum points and inflection
> The family of functions f (t) = C(e-at – e-bt), where a, b, and C are positive numbers and b . a, has been used to model the concentration of a drug injected into the bloodstream at time t = 0. Graph several members of this family. What do they have in c
> Describe how the graph of f varies as c varies. Graph several members of the family to illustrate the trends that you discover. In particular, you should investigate how maximum and minimum points and inflection points move when c changes. You should als
> Describe how the graph of f varies as c varies. Graph several members of the family to illustrate the trends that you discover. In particular, you should investigate how maximum and minimum points and inflection points move when c changes. You should als
> Find dy/dx by implicit differentiation. x2 - 4xy + y2 = 4
> In Example 4 we considered a member of the family of functions f (x) = sin(x + sin cx) that occur in FM synthesis. Here we investigate the function with c = 3. Start by graphing f in the viewing rectangle [0, π] by [-1.2, 1.2]. How many local maximum po
> A company operates 16 oil wells in a designated area. Each pump, on average, extracts 240 barrels of oil daily. The company can add more wells but every added well reduces the average daily ouput of each of the wells by 8 barrels. How many wells should t
> Show that of all the isosceles triangles with a given perimeter, the one with the greatest area is equilateral.
> When John Sullivan was hired as chief investment strategist at the New York headquarters of A. M. Smith Inc., he had indicated that one of his main goals would be to significantly expand the fixed-income unit of the firm’s overall investment portfolio. A
> When Jacqueline and Keith Sommers were “house hunting” five years ago, the mortgage rates were pretty high. The fixed rate on a 30-year mort- gage was 7.25%, while the 15-year fixed rate was at 6.25%. After walking through many homes, they finally reache
> “Boy, this is all so confusing,” said Jason as he stared at the papers on his desk. “If only I had taken the advice of my finance instructor, I would not be in such a predicament today.” Jason Welch, aged 27, graduated five years ago with a degree in foo
> State-sponsored lotteries are extremely popular and highly successful methods by which state governments in many countries raise much-needed funds for financing public expenses, especially education. In Michigan alone, during the year 2015, Michigan Lott
> “Greg, the board of directors’ meeting is scheduled two weeks from today, and I’m depending on you to come up with a realistic and honest appraisal of our company’s position,” said Warren, to his assistant Greg Chapman. “I’m sure that there’s more to us
> “Numbers! I need to see numbers!” exclaimed Marcus in response to com- ments made by the assistant vice-president of Finance, Jeff Smith. Marcus Lenovo, president and chief executive officer of Duralex Inc., had been instrumental in significantly increas
> Andy Gillian, the owner of Gillian Pool & Spa Supplies, paced up and down the balcony of his luxurious Victorian home, overlooking a beautiful backyard, which housed a full-size pool/spa and a sprawling, luscious, green lawn. What was worrying Andy w
> The Ultra Cable Corporation, headquartered in Chicago, Illinois, had thus far enjoyed a fairly steady run-up in revenues and profits. Two years ago, it hired Ron Swenson away from the competition to assist the president, Tom Gray, in navigating the compa
> “It’s amazing how much difference there is in the way proposals are presented at two different firms,” said Art Monk to his assistant, Russell Jacobs, as he pointed to the stack of capital investment proposals piled on his desk. “We sure have our work cu
> Howard Cook could hardly wait to get back to his corporate headquarters in Philadelphia, Pennsylvania, and share all the useful information that he had gathered on performance evaluation with his colleagues in the finance department. Howard was vice pres
> As Cecil shuffled through the stack of files on his desk and clicked away on his mouse, his mind kept racing back to what Jason, his boss, had said to him at the last budget meeting. “We can only fund two or three new projects over the next year,” he sai
> Richard Ponting could feel the pressure as he walked into the executive boardroom with his briefcase containing the data and slides pertaining to his latest proposal. The last couple of years had not been very good for him. Two of the projects that he ha
> When Jenny opened her full-service salon and day spa three years ago, she knew that she would have to make some difficult choices regarding the hiring and firing of qualified professionals such as cosmetologists, estheticians, nail technicians, and massa
> If there was one thing the folks at Creative Products Corporation (CPC) knew well, it was how to come up with useful and unique products in the midst of economic adversity. With current year revenues considerably lower and profit margins shrinking due
> When Wayne concocted his cleaning compound some 20 years ago, all that his wife, Corrine, and he were trying to do was to come up with a sweeter, gentler yet tougher, cleaning product. Little did he realize that someday he would be the proud owner of a m
> Alex Peterson, president of Peterson Investments Inc., was faced with a major decision. One of the firms that his company had invested in, Durawear Corporation, was under severe financial distress. It had therefore sent out a proposal for reorganization,
> It was late Sunday night, and Jassir Amor was getting weary. The big presentation was set for 8 am the next day, and Jassir kept remembering what Greg LeBlanc, the chairman of the mergers and acquisitions (M&A) committee had said to him: “The board membe
> It was a hot, humid afternoon in April. Kirk could feel the pressure mounting. The memo on his desk read, “Please see me immediately!” Kirk knew that, sooner or later, his boss, Brian Daltrey, was going to ask him to implement some quick remedies to impr
> “We better get started on this report,” said Hal to his assistant, Roger, as he straightened up in his seat and fired up his laptop computer. “I’m sure Henry will expect a detailed analysis first thing Monday morning.” Hal Burton, vice president for Inte
> “What do you mean we’ve used up all our cash and lines of credit? I don’t get it! I thought we had a healthy financial position as per last year’s financial statements. How could this have happened, Patrick?” said Donald Hunt to his chief accountant. “If
> It was the end of the fourth quarter. The financial statements had been prepared and circulated to the directors of NuSkin Products Inc. (Tables 2 and 3). The firm’s revenues had surpassed the previous quarter’s revenues by over 20%, and the annual sales
> What is the equivalent of native advertising in commercially released movies?
> Why do you think officers and directors of accused companies accept dissent decrees rather than contest accusations at trial?
