Math 241 Homework 9 Solutions

Section 4.5 (Pages 224-230)

Problem 2. Show that among all rectangles with an 8-m perimeter, the one with largest area is asquare.

Solution The area of a rectangle with side lengths x and y is given by A = xy. We are given

2x + 2y = 8⇔ y = 4 − x⇒ A = 4x − x2

Differentiating with respect to x we have

dA

dx= 4 − 2x

dA

dx= 0⇔ x = 2

Testing the intervals we have

2

Thus the maximum area happens at x = 2 and y = 4 − 2 = 2 and thus the rectangle is a square.

Problem 4. A rectangle has its base on the x-axis and its upper two vertices on the parabolay = 12 − x2. What is the largest area the rectangle can have, and what are its dimensions?

Solution The area of the rectangle is given by A = 2xy. Since we are given that y = 12 − x2 wehave

A = 2x(12 − x2) = 24x − 2x3

Taking the derivative we have

dA

dx= 24 − 6x2 and

dA

dx= 0⇔ x = ±2

Since the length has to be positive we have one critical point of x = 2. Testing we have

2

Thus the maximum area is A = 4(12 − 4) = 32 and happens when x = 2 and y = 12 − 4 = 8

1

Problem 8. A 216 m2 rectangular pea patch is to be enclosed by a fence and divided into two equalparts by another fence parallel to the one of the sides. What dimensions for the outer rectangle willrequire the smallest total length of fence? How much fence will be needed?

Solution The picture is as follows

We are given that 216 = xy⇔ y = 216

xand we want to minimize

P = 3x + 2y = 3x + 432

x

Taking the derivative we havedP

dx= 3 − 432

x2= 3x2 − 432

x2

dP

dxis undefined at x = 0 and

dP

dx= 0⇔ x2 = 144⇒ x = ±12

Since x must be positive we have one critical point of x = 12. Testing the intervals we have that

this critical point is a minimum and thus the dimensions needed are x = 12m and y = 216

12= 18m

2

Problem 12. Find the volume of the largest right circular cone that can be inscribed in a sphereof radius 3.

Mathematical ApplicationsWhenever you are maximizing or minimizing a function of a single vari-able, we urge you to graph it over the domain that is appropriate to the problem you are solving. The graph will provide insight before you cal-culate and will furnish a visual context for understanding your answer.

1. Minimizing perimeter What is the smallest perimeter possible for a rectangle whose area is 16 in2, and what are its dimensions?

2. Show that among all rectangles with an 8-m perimeter, the one with largest area is a square.

3. The figure shows a rectangle inscribed in an isosceles right trian-gle whose hypotenuse is 2 units long.

a. Express the y-coordinate of P in terms of x. (Hint: Write an equation for the line AB.)

b. Express the area of the rectangle in terms of x.

c. What is the largest area the rectangle can have, and what are its dimensions?

x

y

0 1

B

Ax−1

P(x, ?)

4. A rectangle has its base on the x-axis and its upper two vertices on the parabola y = 12 - x2. What is the largest area the rectan-gle can have, and what are its dimensions?

5. You are planning to make an open rectangular box from an 8-in.-by-15-in. piece of cardboard by cutting congruent squares from the cor-ners and folding up the sides. What are the dimensions of the box of largest volume you can make this way, and what is its volume?

6. You are planning to close off a corner of the first quadrant with a line segment 20 units long running from (a, 0) to (0, b). Show that the area of the triangle enclosed by the segment is largest when a = b.

7. The best fencing plan A rectangular plot of farmland will be bounded on one side by a river and on the other three sides by a single-strand electric fence. With 800 m of wire at your disposal, what is the largest area you can enclose, and what are its dimensions?

8. The shortest fence A 216 m2 rectangular pea patch is to be enclosed by a fence and divided into two equal parts by another fence parallel to one of the sides. What dimensions for the outer rectangle will require the smallest total length of fence? How much fence will be needed?

9. Designing a tank Your iron works has contracted to design and build a 500 ft3, square-based, open-top, rectangular steel holding tank for a paper company. The tank is to be made by welding thin stainless steel plates together along their edges. As the production

engineer, your job is to find dimensions for the base and height that will make the tank weigh as little as possible.

a. What dimensions do you tell the shop to use?

b. Briefly describe how you took weight into account.

10. Catching rainwater A 1125 ft3 open-top rectangular tank with a square base x ft on a side and y ft deep is to be built with its top flush with the ground to catch runoff water. The costs associated with the tank involve not only the material from which the tank is made but also an excavation charge proportional to the product xy.

a. If the total cost is

c = 5(x2 + 4xy) + 10xy,

what values of x and y will minimize it?

b. Give a possible scenario for the cost function in part (a).

11. Designing a poster You are designing a rectangular poster to contain 50 in2 of printing with a 4-in. margin at the top and bot-tom and a 2-in. margin at each side. What overall dimensions will minimize the amount of paper used?

12. Find the volume of the largest right circular cone that can be inscribed in a sphere of radius 3.

y

x

3

3

13. Two sides of a triangle have lengths a and b, and the angle between them is u. What value of u will maximize the triangle’s area? (Hint: A = (1>2)ab sin u.)

14. Designing a can What are the dimensions of the lightest open-top right circular cylindrical can that will hold a volume of 1000 cm3? Compare the result here with the result in Example 2.

15. Designing a can You are designing a 1000 cm3 right circular cylindrical can whose manufacture will take waste into account. There is no waste in cutting the aluminum for the side, but the top and bottom of radius r will be cut from squares that measure 2r units on a side. The total amount of aluminum used up by the can will therefore be

A = 8r2 + 2prh

rather than the A = 2pr2 + 2prh in Example 2. In Example 2, the ratio of h to r for the most economical can was 2 to 1. What is the ratio now?

16. Designing a box with a lid A piece of cardboard measures 10 in. by 15 in. Two equal squares are removed from the corners of a 10-in. side as shown in the figure. Two equal rectangles are removed from the other corners so that the tabs can be folded to form a rectangular box with lid.

T

Exercises 4.6

4.6  Applied Optimization 255

Solution We have the volume of the cone is

V = 1

3πr2h = 1

3πx2(y + 3)

We also have thatx2 + y2 = 32 ⇒ x2 = 9 − y2

Using this substitution we have

V = 1

3π(9 − y2)(y + 3) = π

3(9y + 27 − y3 − 3y2)

Thus we havedV

dy= π

3(9 − 3y2 − 6y) = −π(y2 + 2y − 3) = −π(y + 3)(y − 1)

Thus there are critical points at y = −3 and y = 1. Since y must be positive we have y = 1 is theonly valid critical point. Testing we have that dV /dt switches from positive to negative at y = 1 sothere is a relative and thus absolute max at y = 1. Thus gives

V = 1

3π(9 − 1)(1 + 3) = 32π

3

3

Problem 14. What are the dimensions of the lightest open-top right circular cylindrical can thatwill hold a volume of 1000 cm3? Compare the result here with the result in Example 2.

Solution We are given

V = πr2h = 1000⇒ h = 1000

πr2

We want to use the least amount of material and we have

A = πr2 + 2πrh = πr2 + 2πr1000

πr2= πr2 + 2000

r

Differentiating we havedA

dr= 2πr − 2000

r2= 2πr3 − 2000

r2

dA

dr= 0⇔ πr3 = 1000⇔ r = 3

√1000

π= 10

3

√1

π= 10

13√π

We also have that dA/dr is undefined at r = 0 but since r is positive we only have one critical point.Since dA/dr changes from positive to negative at r = 10 3

√πcm we have that there is a relative and

thus absolute max at r = 10 3√π. This gives

h = 1000

π(10 3√

1/π)2= 10π2/3

π= 10

3√π

cm

Problem 18. A rectangle is to be inscribed under the arch of the curve y = 4 cos(0.5x) from x = −πto x = π. What are the dimensions of the rectangle with largest area, and what is the largest area?

Solution DONT DO THIS

4

Problem 20. (a) The U.S. Postal Service will accept a box for domestic shipment only if thesum of its length and girth (distance around) does not exceed 108 in. What dimensions willgive a box with a square end the largest possible volume?

20. a. The U.S. Postal Service will accept a box for domestic ship-ment only if the sum of its length and girth (distance around) does not exceed 108 in. What dimensions will give a box with a square end the largest possible volume?

Square end

Girth = distancearound here

Length

b. Graph the volume of a 108-in. box (length plus girth equals 108 in.) as a function of its length and compare what you see with your answer in part (a).

21. (Continuation of Exercise 20.)

a. Suppose that instead of having a box with square ends you have a box with square sides so that its dimensions are h by h by w and the girth is 2h + 2w. What dimensions will give the box its largest volume now?

w

Girth

h

h

b. Graph the volume as a function of h and compare what you see with your answer in part (a).

22. A window is in the form of a rectangle surmounted by a semicircle. The rectangle is of clear glass, whereas the semicircle is of tinted glass that transmits only half as much light per unit area as clear glass does. The total perimeter is fixed. Find the proportions of the window that will admit the most light. Neglect the thickness of the frame.

23. A silo (base not included) is to be constructed in the form of a cylinder surmounted by a hemisphere. The cost of construction per square unit of surface area is twice as great for the hemisphere as it is for the cylindrical sidewall. Determine the dimensions to be used if the volume is fixed and the cost of construction is to be kept to a minimum. Neglect the thickness of the silo and waste in construction.

T

T

10″

xx

x

x x

x

15″

Base Lid

x x

NO

T T

O S

CA

LE

a. Write a formula V(x) for the volume of the box.

b. Find the domain of V for the problem situation and graph V over this domain.

c. Use a graphical method to find the maximum volume and the value of x that gives it.

d. Confirm your result in part (c) analytically.

17. Designing a suitcase A 24-in.-by-36-in. sheet of cardboard is folded in half to form a 24-in.-by-18-in. rectangle as shown in the accompanying figure. Then four congruent squares of side length x are cut from the corners of the folded rectangle. The sheet is unfolded, and the six tabs are folded up to form a box with sides and a lid.

a. Write a formula V(x) for the volume of the box.

b. Find the domain of V for the problem situation and graph V over this domain.

c. Use a graphical method to find the maximum volume and the value of x that gives it.

d. Confirm your result in part (c) analytically.

e. Find a value of x that yields a volume of 1120 in3.

f. Write a paragraph describing the issues that arise in part (b).

24″

36″

x

24″

x

x x

x x

x x

18″

24″

36″

Base

The sheet is then unfolded.

18. A rectangle is to be inscribed under the arch of the curve y = 4 cos (0.5x) from x = -p to x = p. What are the dimen-sions of the rectangle with largest area, and what is the largest area?

19. Find the dimensions of a right circular cylinder of maximum vol-ume that can be inscribed in a sphere of radius 10 cm. What is the maximum volume?

T

256 Chapter 4: Applications of Derivatives

(b) Graph the volume of a 108-in. box (length plus girth equals 108 in.) of its length and comparewhat you see with your answer in part (a).

Solution

(a) We want to maximize the volume of the box so we can assume that we use all of the 108 in.This gives

4x + y = 108⇒ y = 108 − 4x

We want to maximize the volume which is given by

V = x2y = x2(108 − 4x) = 108x2 − 4x3

Differentiating with respect to x gives

dV

dx= 216x − 12x2 = 12x(18 − x)

Thus there are critical points of x = 0 and x = 18. If we use all of the 108in on the lengththen x = 0 and if we use all of the 108 on the girth then x = 108/4 = 27. Thus our domain forx is [0,27] so we have

V (0) = 0

V (27) = 0

V (18) = 182(108 − 4(18))= 324(36)= 11,664

Thus the volume is maximized when x = 18 inches and y = 108 − 4(18) = 36 inches.

(b)

4x + y = 108⇒ x = 108 − y4

5

Thus we have

V = x2y = (108 − y4

)2

y

Graphing this gives

Which has a maximum at around y = 36, matching our answer from (a).

6

Problem 26. Compare the answers to the following two construction problems.

(a) A rectangular sheet of perimeter 36 cm and dimensions x cm by y cm is to be rolled into acylinder as shown in part (a) of the figure. What values of x and y give the largest volume?

(b) The same sheet is to be revolved about one of the sides of length y to sweep out the cylinderas shown in part (b) of the figure. What values of x and y give the largest volume?

27. Constructing cones A right triangle whose hypotenuse is 23 m long is revolved about one of its legs to generate a right circular cone. Find the radius, height, and volume of the cone of greatest volume that can be made this way.

h

r

"

3

28. Find the point on the line xa +yb

= 1 that is closest to the origin.

29. Find a positive number for which the sum of it and its reciprocal is the smallest (least) possible.

30. Find a positive number for which the sum of its reciprocal and four times its square is the smallest possible.

31. A wire b m long is cut into two pieces. One piece is bent into an equilateral triangle and the other is bent into a circle. If the sum of the areas enclosed by each part is a minimum, what is the length of each part?

32. Answer Exercise 31 if one piece is bent into a square and the other into a circle.

33. Determine the dimensions of the rectangle of largest area that can be inscribed in the right tri-angle shown in the accompanying figure.

34. Determine the dimensions of the rect-angle of largest area that can be inscribed in a semicircle of radius 3. (See accompanying figure.)

35. What value of a makes ƒ(x) = x2 + (a>x) have

a. a local minimum at x = 2?

b. a point of inflection at x = 1?

36. What values of a and b make ƒ(x) = x3 + ax2 + bx have

a. a local maximum at x = -1 and a local minimum at x = 3?

b. a local minimum at x = 4 and a point of inflection at x = 1?

Physical Applications37. Vertical motion The height above ground of an object moving

vertically is given by

s = -16t2 + 96t + 112,

with s in feet and t in seconds. Find

a. the object’s velocity when t = 0;

b. its maximum height and when it occurs;

c. its velocity when s = 0.

38. Quickest route Jane is 2 mi offshore in a boat and wishes to reach a coastal village 6 mi down a straight shoreline from the point nearest the boat. She can row 2 mph and can walk 5 mph. Where should she land her boat to reach the village in the least amount of time?

24. The trough in the figure is to be made to the dimensions shown. Only the angle u can be varied. What value of u will maximize the trough’s volume?

uu

20′

1′

1′

1′

25. Paper folding A rectangular sheet of 8.5-in.-by-11-in. paper is placed on a flat surface. One of the corners is placed on the oppo-site longer edge, as shown in the figure, and held there as the paper is smoothed flat. The problem is to make the length of the crease as small as possible. Call the length L. Try it with paper.

a. Show that L2 = 2x3>(2x - 8.5).

b. What value of x minimizes L2?

c. What is the minimum value of L?

Crease

D C

BPAx

x

L

R

Q (originally at A)"

L2 − x2

26. Constructing cylinders Compare the answers to the following two construction problems.

a. A rectangular sheet of perimeter 36 cm and dimensions x cm by y cm is to be rolled into a cylinder as shown in part (a) of the figure. What values of x and y give the largest volume?

b. The same sheet is to be revolved about one of the sides of length y to sweep out the cylinder as shown in part (b) of the figure. What values of x and y give the largest volume?

x

y

y

(a)

Circumference = xy

x

(b)

r = 3

w

h

4

3

5w

h

4.6  Applied Optimization 257

Solution

(a) We know that2x + 2y = 36⇔ 2y = 36 − 2x⇔ y = 18 − x

Andx = 2πr⇔ r = x

2π

And we want to maximize

V = πr2 = π ( x2π

)2

(18 − x) = π ( x2

4π2)(18 − x) = 9

2πx2 − 1

4πx3

Differentiating with respect to x gives

dV

dx= 9

πx − 3

4πx2 = 3

πx(3 − x

4)

This gives critical numbers x = 0 and x = 12. Since the domain for x is [0,18] we have

V (0) = 0

V (18) = 0

V (12) = 144π

4π2(6)

= 216

π

Thus the maximum volume occurs when x = 12 and y = 18 − 12 = 6

7

(b) Since we are using the same sheet we still have y = 18 − x. We want to maximize the volume

V = πx2y = πx(18 − x) = 18πx2 − πx3

Taking the derivative we have

dV

dx= 36πx − 3πx2 = 3πx(12 − x)

This gives to critical points x = 0 and x = 12. We have the same domain of [0,18].

V (0) = 0

V (18) = 0

V (12) = π144(6) = 864π

Thus the maximum volume occurs when x = 12 and y = 6

Problem 48. Suppose that c(x) = x3 − 20x2 + 20,000x is the cost of manufacturing x items. Finda production level that will minimize the average cost of making x items.

Solution The average cost is given by

c(x)x= x2 − 40x + 20,000

Differentiating this we getdc

dx= 2x − 40 = 2(x − 20)

Which gives a critical number of x = 20 items. Since the average cost is a parabola we have thatx = 20 items gives the minimized average cost.

8

Section 4.6 (Pages 233-234)

Problem 2. Use Newton’s method to estimate the one real solution of x3 + 3x + 1 = 0. Start withx0 = 0 then find x2.

Solutionf(x) = x3 + 3x + 1⇒ f ′(x) = 3x2 + 3

x1 = 0 − f(0)f ′(0)

= −1

3

x2 = −1

3− f(−1/3)f ′(−1/3)

= −1

3− (−1/27 − 1 + 1)

(3(1/9) + 3)

= −1

3+ 1/27

10/3

= −1

3+ 1

90

= −29

90

9

Problem 6. Use Newton’s method to find the negative fourth root of 2 by solving the equationx4 − 2 = 0. Start with x0 = −1 and find x2.

Solutionf(x) = x4 − 2⇒ f ′(x) = 4x3

x1 = −1 − f(−1)f ′(−1)

= −1 − −1

−4

= 1 − 1

4

= 3

4

x2 = 3

4− f(3/4)f ′(3/4)

= 3

4− (81/256) − 2

(108/64)

= 3

4− 81 − 512

432

= 3

4+ 431

432

= 324 + 431

432

= 755

432

10

Section 4.7 (Pages 241-244)

Problem 6. Find an antiderivative for each function. Do as many as you can mentally. Checkyour answers by differentiation:

(a) − 2

x3

(b)1

2x3

(c) x3 − 1

x3

Solution

(a) x−2 has derivative −2x−3 = − 2

x3so any equation of the form x−2 +C is an antiderivative.

(b) −1

4x−2 has derivative

1

2x−3 = 1

2x3so any equation of the form −1

4x−2 +C is an antiderivative.

(c)1

4x4+1

2x−2 has derivative x3− 1

x3so any equation of the form

1

4x4+1

2x−2+C is an antiderivative.

11

Problem 8. Find an antiderivative for each function. Do as many as you can mentally. Checkyour answers by differentiation:

(a)4

33√x

(b)1

3 3√x

(c) 3√x + 1

3√x

Solution

(a)4

33√x = 4

3x1/3

So the general antiderivative can be found by

4

3⋅ x

1/3+1

1/3 + 1+C = 4

3⋅ x

4/3

4/3 +C = x4/3 +C

(b)1

3 3√x= 1

3x1/3

So the general antiderivative can be found by

1

3⋅ x

1/3+1

1/3 + 1+C = 1

3⋅ x

4/3

4/3 +C =1

4x4/3 +C

(c)

3√x + 1

3√x= x1/3 + x−1/3

So the general antiderivative can be found by

x1/3+1

1/3 + 1+ x−1/3+1

−1/3 + 1+C = x

4/3

4/3 +x2/3

2/3 +C =3

4x4/3 + 3

2x2/3 +C

12