Optimization

Optimization
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3.7

1. Suppose some friends complain to you that they can't work any of the problems in this section. When you ask to see their work, they say that they couldn't even get started. In the text, we have emphasized sketching a picture and defining variables. Part of the benefit of this is to help you get started writing something (anything) down. Do you think this advice helps? What do you think is the most difficult aspect of these problems? Give your friends the best advice you can.

2. We have neglected one important aspect of optimization problems, an aspect that might be called “common sense. ” For example, suppose you are finding the optimal dimensions for a fence and the mathematical solution is to build a square fence of length feet on each side. At the meeting with the carpenter who is going to build the fence, what length fence do you order? Why is probably not the best way to express the length? We can approximate . What would you tell the carpenter? Suppose the carpenter accepts measurements down to the inch. Assuming that the building constraint was that the perimeter of the fence could not exceed a certain figure, why should you truncate to 22 '4 '' instead of rounding up to 22 '5 ''?

3. In Example 7.3, we stated that is minimized by exactly the same x-value(s) as f (x). Use the fact that is an increasing function to explain why this is true.

4. Suppose that f (x) is a continuous function with a single critical number, and f (x) has a local minimum at that critical number. Explain why f (x) also has an absolute minimum at the critical number.

5. Give an example showing that f (x) and sin(f (x)) need not be minimized by the same x-values.

6. True or false: e^{f (x)} is minimized by exactly the same x - value(s) as f (x) .

7. A three-sided fence is to be built next to a straight section of river, which forms the fourth side of a rectangular region. The enclosed area is to equal 1800 ft². Find the minimum perimeter and the dimensions of the corresponding enclosure.

8. A three-sided fence is to be built next to a straight section of river, which forms the fourth side of a rectangular region. There is 96 feet of fencing available. Find the maximum enclosed area and the dimensions of the corresponding enclosure.

9. A two-pen corral is to be built. The outline of the corral forms two identical adjoining rectangles. If there is 120 ft of fencing available, what dimensions of the corral will maximize the enclosed area?

10. A showroom for a department store is to be rectangular with walls on three sides, 6 - ft door openings on the two facing sides and a 10 - ft door opening on the remaining wall. The showroom is to have 800 ft² of floor space. What dimensions will minimize the length of wall used?

11. Show that the rectangle of maximum area for a given perimeter P is always a square.

12. Show that the rectangle of minimum perimeter for a given area A is always a square.

13. Find the point on the curve y = x² closest to the point (0, 1).

14. Find the point on the curve y = x² closest to the point (3, 4).

15. Find the point on the curve y = cos x closest to the point (0, 0).

16. Find the point on the curve y = cos x closest to the point (1, 1).

17. In exercises 13 and 14, find the slope of the line through the given point and the closest point on the given curve. Show that, in each case, this line is perpendicular to the tangent line to the curve at the given point.

18. Sketch the graph of some function y = f (x) and mark a point not on the curve. Explain why the result of exercise 17 is true. (Hint: Pick a point for which the joining line is not perpendicular and explain why you can get closer.)

19. A box with no top is to be built by taking a 6'' - by - 10'' sheet of cardboard and cutting x - in. squares out of each corner and folding up the sides. Find the value of x that maximizes the volume of the box.

20. A box with no top is to be built by taking a 12'' - by - 16'' sheet of cardboard and cutting x - in. squares out of each corner and folding up the sides. Find the value of x that maximizes the volume of the box.

21. A water line runs east-west. A town wants to connect two new housing developments to the line by running lines from a single point on the existing line to the two developments. One development is 3 miles south of the existing line, the other development is 4 miles south of the existing line and 5 miles east of the first development. Find the place on the existing line to make the connection to minimize the total length of new line.

22. A company needs to run an oil pipeline from an oil rig 25 miles out to sea to a storage tank that is 5 miles inland. The shoreline runs east-west and the tank is 8 miles east of the rig. Assume it costs $ 50 thousand dollars per mile to construct the pipeline underwater and $ 20 thousand dollars per mile to construct the pipeline on land. The pipeline will be built in a straight line from the rig to a selected point on the shoreline, then in a straight line to the storage tank. What point on the shoreline should be selected to minimize the total cost of the pipeline?

23. A city wants to build a new section of highway to link an existing bridge with an existing highway interchange, which lies 8 miles to the east and 10 miles to the south of the bridge. The first 4 miles south of the bridge is marsh land. Assume that the highway costs $ 5 million dollars per mile over marsh and $ 2 million dollars per mile over dry land. The highway will be built in a straight line from the bridge to the edge of the marsh, then in a straight line to the existing interchange. At what point should the highway emerge from the marsh in order to minimize the total cost of the new highway? How much is saved over building the new highway in a straight line from the bridge to the interchange? (Hint: Use similar triangles to find the point on the boundary corresponding to a straight path and evaluate your cost function at that point.)

24. After construction has begun on the highway in exercise 23, the cost per mile over marsh land is re-estimated at $ 6 million dollars. Find the point on the marsh/dry land boundary that would minimize the total cost of the highway with the new cost function. If the construction is too far along to change paths, how much extra cost is there in using the path from exercise 23?

25. After construction has begun on the highway in exercise 23, the cost per mile over dry land is reestimated at $ 3 million dollars. Find the point on the marsh/dry land boundary that would minimize the total cost of the highway with the new cost function. If the construction is too far along to change paths, how much extra cost is there in using the path from exercise 23?

26. In an endurance contest, contestants 2 miles at sea need to reach a location 2 miles inland and 3 miles east (the shoreline runs east-west). Assume a contestant can swim 4 mph and run 10 mph. To what point on the shoreline should the person swim to minimize the total time? Compare the amount of time spent in the water and the amount of time spent on land.

27. Suppose that light travels from point A to point B as shown in the figure. (Recall that light always follows the path that minimizes time.) Assume that the velocity of light above the boundary line is v₁ and the velocity of light below the boundary is v₂. Show that the total time to get from point A to point B is

Write out the equation T' (x) = 0, replace the square roots using the sines of the angles in the figure and derive Snell's Law = .

Exercise 27

28. Suppose that light reflects off a mirror to get from point A to point B as indicated in the figure. Assuming a constant velocity of light, we can minimize time by minimizing the distance traveled. Find the point on the mirror that minimizes the distance traveled. Show that the angles in the figure are equal (the angle of incidence equals the angle of reflection).

Exercise 28

29. A soda can is to hold 12 fluid ounces. Suppose that the bottom and top are twice as thick as the sides. Find the dimensions of the can which minimize the amount of material used. (Hint: Instead of minimizing surface area, minimize the cost, which is proportional to the product of the thickness and the area.)

30. Following Example 7.5, we mentioned that real soda cans have a radius of about 1.156''. Show that this radius minimizes the cost if the top and bottom are 2.23 times as thick as the sides.

31. The human cough is intended to increase the flow of air to the lungs, by dislodging any particles blocking the windpipe and changing the radius of the pipe. Suppose a windpipe under no pressure has radius r₀. The velocity of air through the windpipe at radius r is approximately V(r) = cr²(r₀ - r) for some constant c. Find the radius that maximizes the velocity of air through the windpipe. Does this mean the windpipe expands or contracts?

32. To supply blood to all parts of the body, the human artery system must branch repeatedly. Suppose an artery of radius r branches off from an artery of radius R ( R > r ) at an angle . The energy lost due to friction is approximately

Find the value of that minimizes the energy loss.

33. In an electronic device, individual circuits may serve many purposes. In some cases, the flow of electricity must be controlled by reducing the power instead of amplifying it. In the circuit shown below, a voltage V volts and resistance R ohms are given. We want to determine the size of the remaining resistor (x ohms). The power absorbed by the circuit is

p(x) =
Find the value of x that maximizes the power absorbed.

34. In an AC circuit with voltage V(t) = vsin 2 ft, a voltmeter actually shows the average (root-mean-square) voltage of . If the frequency is f = 60 (Hz) and the meter registers 115 volts, find the maximum voltage reached. [Hint: this is “obvious” if you determine v and think about the graph of V(t) .]

35. A Norman window has the outline of a semicircle on top of a rectangle, as shown below. Suppose there is 8 + feet of wood trim available. Discuss why a window designer might want to maximize the area of the window. Find the dimensions of the rectangle (and, hence, the semicircle) that will maximize the area of the window.

36. Suppose a wire 2 ft long is to be cut into two pieces, each of which will be formed into a square. Find the size of each piece to maximize the total area of the two squares.

37. An advertisement consists of a rectangular printed region plus 1 - in. margins on the sides and 2 - in. margins at top and bottom. Ifthe area of the printed region is to be 92 in², find the dimensions of the printed region and overall advertisement that minimize the total area?

38. An advertisement consists of a rectangular printed region plus 1 - in. margins on the sides and 1.5 - in. margins at top and bottom. If the total area of the advertisement is to be 120 in², what dimensions should the advertisement be to maximize the area of the printed region?

39. A farmer relishes growing cucumbers. A government subsidy is available to growers of more than 2 acres of cucumbers. The farmer has 10 acres available to plant. The net income for the farmer planting A acres is 2A³ - 33A² + 108A -310 dollars. Determine the best strategy for the farmer.

40. The owners of the Big Belly Deli run a special on bologna sandwiches, with a limit of six per customer. The amount of heartburn obtained from n sandwiches is h(n) = n³ + 12n -11 gasu's (gastronomical units). How many sandwiches gives the worst case of heartburn?

41. In sports where balls are thrown or hit, the ball often finishes at a different height than it starts. Examples include a downhill golf shot and a basketball shot. In the diagram, a ball is released at an angle and finishes at an angle above the horizontal (for downhill trajectories, would be negative). Neglecting air resistance and spin, the horizontal range is given by if the initial velocity is v and g is the gravitational constant. In the following cases, find to maximize R (treat v and g as constants): (a) = 10°, (b) = 0° and (c) = - 10°. Verify that = 45° + °/2 maximizes the range.

42. For your favorite sport in which it is important to throw or hit a ball a long way, explain the result of exercise 41 in the language of your sport.

43. A ball is thrown from s = b to s = a (where a < b ) with initial speed v₀. Assuming that air resistance is proportional to speed, the time it takes the ball to reach s = a is where c is a constant of proportionality. A baseball player is 300ft from home plate and throws a ball directly toward home plate with an initial speed of 125 ft/s. Suppose that c = 0.1. How long does it take the ball to reach home plate? Another player standing x feet from home plate has the option of catching the ball and then, after a delay of 0.1 s, relaying the ball toward home plate with an initial speed of 125 ft/s. Find x to minimize the total time for the ball to reach home plate. Is the straight throw or the relay faster? What, if anything, changes if the delay is 0.2 s instead of 0.1 s?

44. For the situation in exercise 43, for what length delay is it equally fast to have a relay and not have a relay? Do you think that you could catch and throw a ball in such a short time? Why do you think it is considered important to have a relay option in baseball?

45. Repeat exercises 43 and 44 if the second player throws the ball with initial speed 100 ft/s.

46. For a delay of 0.1s in exercise 43, find the value of the initial speed of the second player's throw for which it is equally fast to have a relay and not have a relay.

47. In exercise 64 in section 3.3, you did a preliminary investigation of Kepler's wine cask problem. You showed that a height-to-diameter ratio ( x/y ) of for a cylindrical barrel will maximize the volume (see Figure a). However, real wine casks are bowed out (like beer kegs). Kepler continued his investigation of wine cask construction by approximating a cask with the straight-sided barrel in Figure b. It can be shown (we told you Kepler was good!) that the volume of this barrel is . Treating w and z as constants, show that V'(y) = 0 if y = w/2. Recall that such a critical point can correspond to a maximum or minimum of V(y), but it also could correspond to something else (e.g., inflection point). To discover which one we have here, redraw Figure b to scale (show the correct relationship between 2y and w ). In physical terms (think about increasing and decreasing y ), argue that this critical point is neither a maximum nor minimum. Interestingly enough, such a nonextreme critical point would have a definite advantage to the Austrian vintners. Recall that their goal was to convert the measurement z into an estimate of the volume. The vintners would hope that small imperfections in the dimensions of the cask would have little effect on the volume. Explain why V' (y) = 0 means that small variations in y would convert to small errors in the volume V.

Figure a
Figure b

48. The following problem is fictitious, but involves the kind of ambiguity that can make technical jobs challenging. The Band Candy Company decides to liquidate one of its candies. The company has 600,000 bags in inventory that it wants to sell. The candy had cost 35 cents per bag to manufacture and originally sold for 90 cents per bag. A marketing study indicates that if the candy is priced at p cents per bag, approximately Q(p) = - p² + 40p + 250 thousand bags will be sold. Your task as consultant is to recommend the best selling price for the candy. As such, you should do the following: (a) find p to maximize Q(p) ; (b) find p to maximize 10pQ(p), which is the actual revenue brought in by selling the candy. Then form your opinion, based on your evaluation of the relative importance of getting rid of as much candy as possible and making the most money possible.