MA005: Calculus I

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 $P$ 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 $\mathrm{P}$ 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 $P$ 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 $P$ feet of fencing if one edge of the garden borders a river and does not need to be fenced?

3. You have $120$ feet of fencing to construct a pen with $4$ equal sized stalls.
(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?

Fig. 8

(b) The square pen in Fig. 9 uses $120$ feet of fencing and encloses a larger area ($400$ square feet) than the best design in part (a). Design a pen which uses only $120$ feet of fencing and has $4$ equal sized stalls but which encloses even more than $400$ square feet. (Suggestion: don't use rectangles and squares.)

Fig. 9

5. You have a $10$ inch by $10$ inch piece of cardboard which you plan to cut and fold as shown in Fig. 11 to form a box with a top. Find the dimensions of the box which has the largest volume.

Fig. 11

7. (a) Determine the dimensions of the least expensive cylindrical can which will hold $100$ cubic inches if the materials cost $2¢, 5¢$ and $3¢$ respectively for the top, bottom and sides.
(b) How do the dimensions of the least expensive can change if the bottom material costs more than $5¢$ 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 $8$ meters per second and swim about $2 \mathrm{~m} / \mathrm{s}$, how far along the shore should you run before diving into the water in order to reach the swimmer as quickly as possible?

Fig. 13

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 $\$ 3000$ per mile and to Denton it costs $\$ 7000$ per mile.

Fig. 15

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.

Fig. 17

15. Two sides of a triangle are $7$ and $10$ inches respectively. What is the length of the third side so the area of the triangle will be greatest? (This problem can be done without using calculus. How? If you do use calculus, consider the angle $\theta$ between the two sides.)

17. Find the dimensions of the rectangle with the largest area if the base must be on the $\mathrm{x}$-axis and its other two corners are on the graph of
(a) $\mathrm{y}=16-\mathrm{x}^{2}$ on $[-4,4]$
(b) $x^{2}+y^{2}=1$ on $[-1,1]$
(c) $|x|+|y|=1$ on $[-1,1]$
(d) $\mathrm{y}=\cos (\mathrm{x})$ on $[-\pi / 2, \pi / 2]$

19. You have a long piece of $12$ inch wide metal which you are going to fold along the center line to form a V-shaped gutter (Fig. 20). What angle $\theta$ will give the gutter which holds the most water (the largest cross-sectional area)?

Fig. 20

21. You have a $6$ inch diameter circle of paper which you want to form into a drinking cup by removing a pie-shaped wedge and forming the remaining paper into a cone (Fig. 22). Find the height and top radius of the cone so the volume of the cup is as large as possible.

Fig. 22

23. You own a small airplane which holds a maximum of $20$ passengers. It costs you $\$ 100$ per flight from St. Thomas to St. Croix for gas and wages plus an additional $\$ 6$ per passenger for the extra gas required by the extra weight. The charge per passenger is $\$ 30$ each if $10$ people charter your plane ( $10$ is the minimum number you will fly), and this charge is reduced by $\$ 1$ per passenger for each passenger over 10 who goes (that is, if $11$ go they each pay $\$ 29$, if $12$ go they each pay $\$ 28$, etc.). What number of passengers on a flight will maximize your profits?

25. Profit is revenue minus expenses. Assume that revenue and expenses are differentiable functions and show that when profit is maximized, then marginal revenue $(\mathrm{d} \mathrm{R} / \mathrm{d} \mathrm{x})$ equals marginal expense $(\mathrm{dE} / \mathrm{dx})$.

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 gravity $(cg)$ of 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 $\mathrm{cg}$ is directly above the bottom rim. (Fig. 25a) Find $\theta$.

(ii) For a partly filled can more thinking was needed. Some ideas you will see in chapter 5 let us calculate that the $\mathrm{cg}$ of a can containing $\mathrm{x}$ $\mathrm{cm}$ of cola is $\mathrm{C}(\mathrm{x})=\frac{360+9.6 \mathrm{x}^{2}}{60+19.2 \mathrm{x}} \mathrm{cm}$ above the bottom of the can. Find the height $\mathrm{x}$ of cola in the can which will make the $\mathrm{cg}$ as low as possible.

(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 $\mathrm{x}$ $\mathrm{cm}$ of cola. (Fig. 25b)

(iv) If the can contained $\mathbf{x}$ $\mathrm{cm}$ of liquid cola, could we tilt it more or less far than the frozen cola before it would fall over?

Fig. 25

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 $x$-axis between $0$ and $1(0 \leq x \leq 1)$ and one corner of the rectangle is on the curve $\mathrm{y}=\mathrm{x}^{2}$ (Fig. 27a). What is the area of this rectangle?
(b) Generalize the problem in part (a) for the parabola $\mathrm{y}=\mathrm{Cx}^{2}$ with $\mathrm{C} > 0$ and $0 \leq \mathrm{x} \leq 1$ (Fig. 27b).
(c) Generalize for the parabola $\mathrm{y}=\mathrm{Cx}^{2}$ with $\mathrm{C} > 0$ and $0 \leq \mathrm{x} \leq \mathrm{B}$ (Fig. 27c).

Fig. 27

31. (a) The base of a right triangle is $50$ and the height is $20$ (Fig. 28a). 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.
(b) The base of a right triangle is $\mathrm{B}$ and the height is $\mathrm{H}$ (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.

Fig. 28

33. Determine the dimensions of the least expensive cylindrical can which will hold $\mathrm{V}$ cubic inches if the top material costs $\$A$ per square inch, the bottom material costs $\$ \mathrm{~B} / \mathrm{in}^{2}$, and the side material costs $\mathrm{\$ C} / \mathrm{in}^{2}$.

Source: Dale Hoffman, https://s3.amazonaws.com/saylordotorg-resources/wwwresources/site/wp-content/uploads/2012/12/MA005-4.5-Applied-Maximum-and-Minimum.pdf
This work is licensed under a Creative Commons Attribution 3.0 License.