Work through the odd-numbered problems 1-33. Once you have completed the problem set, check your answers.
1. (a) You have 200 feet of fencing to enclose a rectangular vegetable garden. What should the dimensions of your garden be in order to enclose the largest area?
(b) Show that if you have feet of fencing available, the garden of greatest area is a square.
(c) What are the dimensions of the largest rectangular garden you can enclose with feet of fencing if one edge of the garden borders a straight river and does not need to be fenced?
(d) Just thinking - calculus will not help with this one: What do you think is the shape of the largest garden which can be enclosed with feet of fencing if we do not require the garden to be rectangular? What do you think is the shape of the largest garden which can be enclosed with feet of fencing if one edge of the garden borders a river and does not need to be fenced?
3. You have
(a) If the pen is rectangular and shaped like the Fig. 8, what are the dimensions of the pen of largest area and what is that area?
(b) The square pen in Fig. 9 uses
5. You have a
7. (a) Determine the dimensions of the least expensive cylindrical can which will hold
(b) How do the dimensions of the least expensive can change if the bottom material costs more than per square inch?
9. You are a lifeguard standing at the edge of the water when you notice a swimmer in trouble (Fig. 13). Assuming you can run about
11. You have been asked to determine where a water works should be built along a river between Chesterville and Denton (see Fig. 15 ) to minimize the total cost of the pipe to the towns.
(a) Assume that the same size (and cost) pipe is used to each town. (This part can be done quickly without using calculus.)
(b) Assume that the pipe to Chesterville costs per mile and to Denton it costs per mile.
13. U.S. postal regulations state that the sum of the length and girth (distance around) of a package must be no more than 108 inches. (Fig. 17)
(a) Find the dimensions of the acceptable box with a square end which has the largest volume.
(b) Find the dimensions of the acceptable box which has the largest volume if its end is a rectangle twice as long as it is wide.
(c) Find the dimensions of the acceptable box with a circular end which has the largest volume.
15. Two sides of a triangle are
17. Find the dimensions of the rectangle with the largest area if the base must be on the
19. You have a long piece of
21. You have a
23. You own a small airplane which holds a maximum of
25. Profit is revenue minus expenses. Assume that revenue and expenses are differentiable functions and show that when profit is maximized, then marginal revenue
27. After the table was wiped and the potato chips dried off, the question remained: "Just how far could a can of cola be tipped before it fell over?"
(i) For a full can or an empty can the answer was easy: the center of gravityof the can is at the middle of the can, half as high as the height of the can, and we can tilt the can until the is directly above the bottom rim. (Fig. 25a) Find .
(ii) For a partly filled can more thinking was needed. Some ideas you will see in chapter 5 let us calculate that the
(iii) Assuming that the cola is frozen solid (so the top of the cola stays parallel to the bottom of the can), how far can we tilt a can containing
(iv) If the can contained
29. (a) Find the dimensions of the rectangle with the greatest area that can be built so the base of the rectangle is on the
(b) Generalize the problem in part (a) for the parabola with and (Fig. 27b).
(c) Generalize for the parabola with and (Fig. 27c).
31. (a) The base of a right triangle is
(b) The base of a right triangle is and the height is (Fig. 28b). Find the dimensions and area of the rectangle with the greatest area that can be enclosed in the triangle if the base of the rectangle must lie on the base of the triangle.
(c) State your general conclusion from part (b) in words.
33. Determine the dimensions of the least expensive cylindrical can which will hold
Source: Dale Hoffman, https://s3.amazonaws.com/saylordotorg-resources/wwwresources/site/wp-content/uploads/2012/12/MA005-4.5-Applied-Maximum-and-Minimum.pdf
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