11. The fanner of Problem 10 decides to make three identi
cal pens with his 80 feet of fence, as shown in Figure 19. What
Figure 18
B
A. fanner has 80 feet of fence with which he plans to en
close a rectangular pen along one side of his 100-foot barn, as
shown in Figure 18 (the side along the barn needs no fence).
What are the dimensions of the pen that has maximum area?
i 10.
9. Find the volume of the largest open box that can be
made from a piece of cardboard 24 inches square by cutting
equal squares from the corners and turning up the sides (see
Example 1).
8. Show that for a rectangle of given perimeter K the one
with maximum area is a square.
7. What number exceeds its square by the maximum
amount? Begin by convincing yourself that this number is on the
interval [0,11.
.
2 that are closest
Find the points on the parabola x = 2y
to the point (10, 0). Hint: Minimize the square of the distance
between (x, v) and (10, 0).
5. Find the points on the parabola y = 2 that are closest
to the point (0, 5). I-tint: Minimize the square of the distance
between (x, y) and (0,5).
H’
14. A farmer wishes to fence off three identical adjoining rec
tangular pens (see Figure 22), each with 300 square feet of area.
What should the width and length of each pen be so that the least
amount of fence is required?
15. Suppose that the outer boundary of the pens in Problem 14
requires heavy fence that costs $3 per foot, but that the two
H
13. A farmer wishes to fence off two identical adjoining rec
tangular pens, each with 900 square feet of area, as shown in
Figure 21. What are x and y so that the least amount of fence is
required?
Figure 20
Barn
12. Suppose that the farmer of Problem 10 has 180 feet of
fence and wants the pen to adjoin to the whole side of the
100-foot barn, as shown in Figure 20. What should the dimensions
40 in this case.
x
be for mnaximumn area? Note that 0
2 +
fri
24. Find the equation of the line that is tangent to the ellipse
y = a
2
a
b in the first quadrant and forms with the coor
2
dinate axes the triangle with smallest possible area (a and b are
positive constants).
25. Find the greatest volume that a right circular cylinder can
have if it is inscribed in a sphere of radius r.
26. Show that the rectangle with maximum perimeter that
can be inscribed in a circle is a square.
27. What are the dimensions of the right circular cylinder
wmth greatest curved surface area that can be inscribed in a
sphere of radius r?
28. The illumination at a point is inversely proportional to
the square of the distance of the point from the light source and
directly proportional to the intensity of the light source. If two
lmght sources are s feet apart and their intensities are ‘m and ‘2’ re
sPectively, at what point between them will the sum of their illu
mmnations be a minimum?
29. A wire of length 100 centimeters is cut into two pieces;
One is bent to form a square, and the other is bent to form an
equi1at triangle. Where should the cut be made if (a) the sum
of the two areas is to be a minimum; (b) a maximum? (Allow the
POSsibility of no cut.)
23. At 7:00 AM. one ship was 60 miles due east from a second
ship. If the first ship sailed west at 20 miles per hour and the sec
ond ship sailed southeast at 30 miles per hour, when were they
closest together?
22. A powerhouse is located on one bank of a straight river
that is w feet wide. A factory is situated on the opposite bank
of the river, L feet downstream from the point A directly oppo
site the powerhouse. What is the most economical path for a
cable connecting the powerhouse to the factory if it costs a dol
lars per foot to lay the cable under water and b dollars per foot
onland(a > b)?
Figure 19
4. Find two numbers whose product is —12 and the sum of
whose squares is a minimum.
20. In Problem 19, suppose that the woman will be picked up
by a car that will average 50 miles per hour when she gets to the
shore. Then where should she land?
21. In Problem 19, suppose that the woman uses a motorboat
that goes 20 miles per hour. Then where should she land?
Ihaamam
dimensions for the total enclosure make the area of the pens as
large as possible?
19. A small island is 2 miles from the nearest point F on the
straight shoreline of a large lake. If a woman on the island can
row a boat 3 miles per hour and can walk 4 miles per hour, where
should the boat be landed in order to arrive at a town 10 miles
down the shore from P in the least time?
18. A right circular cone is to be inscribed in another right
circular cone of given volume, with the same axis and with the
vertex of the inner cone touching the base of the outer cone.
What must be the ratio of their altitudes for the inscribed cone to
have maximum volume?
o
/4,
2
17. Find the points P and Q on the curve y = x
2\/, that are closest to and farthest from the point
(0, 4). flint: The algebra is simpler if you consider the square of
the required distance rather than the distance itself.
internal partitions require fence costing only $2 per foot. What
dimensions x and y will produce the least expensive cost for the
pens?
16. Solve Problem 14 assuming that the area of each pen is
900 square feet. Study the solution to this problem and to Prob
lem 14 and make a conjecture about the ratio of x/y in all prob
lems of this type. Try to prove your conjecture.
3. For what number does the principal fourth root exceed
twice the number by the largest amount?
2. For what number does the principal square root exceed
eight times the number by the largest amount?
1. Find two numbers whose product is —16 and the sum of
whose squares is a minimum
Problem Set 3.4
Note that
To maximize profit, we set dP/dx = 0 and solve. This gives x = 975 as the
oniy critical point to consider. It does provide a maximum, as may be checked by
the First Derivative Test. The maximum profit is P(975) = $1898.25.
174 Chapter 3 Applications of the Derivative
175
sin 2t +
cos 2t
39. Of all rectangles with a given diagonal, find the one with
the maximum area.
38. Find the dimensions of the rectangle of greatest area that
can be inscribed in the ellipse 2
/a + 2
x
/b = 1.
y
37. Of all right circular cylinders with a given surface area,
find the one with the maximum volume. Note: The ends of the
cylinders are closed.
36. A rectangle is to be inscribed in a semicircle of radius r, as
shown in Figure 25. What are the dimensions of the rectangle if
its area is to be maximized?
—
35. A rectangle has two corners on the x-axis and the other
two on the parabola y
12
, with y
2
x
0 (Figure 24). What
are the dimensions of the rectangle of this type with maximum
area?
34. A fence ii feet high runs parallel to a tall building and w
feet from it (Figure 23). Find the length of the shortest ladder
that will reach from the ground across the top of the fence to the
wall of the building.
33. A flower bed will be in the shape of a sector of a circle (a
pie-shaped region) of radius r and vertex angle 0. Find r and 0 if
its area is a constant A and the perimeter is a minimum.
What is the farthest that the weight gets from the origin?
x
32. A weight connected to a spring moves along the x-axis so
that its x-coordinate at time t is
31. An observatory is to be in the form of a right circular
cylinder surmounted by a hemispherical dome. If the hemispher
ical dome costs twice as much per square foot as the cylindrical
wall, what are the most economical proportions for a given
volume?
30. A closed box in the form of a rectangular parallelepiped
with a square base is to have a given volume. If the material used
in the bottom costs 20% more per square inch than the material
in the sides, and the material in the top costs 50% more per
square inch than that of the sides, find the most economical pro
portions for the box.
Section 3.4 Practical Problems
Chapter 3 Applications of the Derivative
Figure 28
iZI
-
45. One corner of a long narrow strip of paper is folded over
so that it just touches the opposite side, as shown in Figure 30.
With parts labeled as indicated, determine x in order to
(a) maximize the area of triangle A;
44. I have enough pure silver to coat 1 square meter of sur
face area. I plan to coat a sphere and a cube. What dimensions
should they be if the total volume of the silvered solids is to be a
maximum? A minimum? (Allow the possibility of all the silver
going onto one solid.)
Figure 29
T
zJ
y
43. A covered box is to be made from a rectangular sheet of
cardboard measuring 5 feet by 8 feet. This is done by cutting out
the shaded regions of Figure 29 and then folding on the dotted
lines. What are the dimensions x, y, and z that maximize the
volume?
42. A huge conical tank is to be made from a circular piece of
sheet metal of radius 10 meters by cutting out a sector with vertex
angle 9 and then welding together the straight edges of the re
niaining piece (Figure 28). Find 6 so that the resulting cone has
the largest possible volume.
Figure 27
41. A metal rain gutter is to have 3-inch sides and a 3-inch
horizontal bottom, the sides making an equal angle 6 with the
bottom (Figure 27). What should 0 be in order to maximize the
0
carrying capacity of the gutter? Note: 0
Figure 26
40. A humidifier uses a rotating disk of radius r, which is par
tially submerged in water. The most evaporation occurs when the
exposed wetted region (shown as the upper shaded region in
Figure 26) is maximized. Show that this happens when h (the dis
tance from the center to the water) is equal to r/\/i.
176
—
Figure 32
=
2 + m
hni(h
2
—
2hm cos
sin 0
=
—
--
—
—
49. The earth’s position in the solar system at time t can
be described approximately by P(93 cos(2v-t), 93 sin(2irt)),
where the sun is at the origin and distances are measured in
millions of miles. Suppose that an asteroid has position
Q(60 cos[2r(1.51t
1)], 120 sin[2n-(1.51t
1)]). When, over
the time period [0, 20] (i.e., over the next 20 years), does the
asteroid come closest to the earth? How close does it come?
L1 Li
(a) Find the position of the object when it is closest to the
observer.
(b) Find the position of the object when it is farthest from the
observer.
x + 100. An observer stands
from the edge of a 100-foot cliff fol
2 feet from the bottom of the cliff.
lows the path given by y
Ei E1 48. An object thrown
(a) For h = 3 and m = 5, determine L’, L, and 4) at the instant
when L’ is largest.
(b) Rework part (a) when h = 5 and m = 13.
(c) Based on parts (a) and (b), make conjectures about the val
ues of L’, L, and 4) at the instant when the tips of the hands
are separating most rapidly.
(d) Try to prove your conjectures.
L’(6)
8)_1/2
F 47. A clock has hour and minute hands of lengths h and in,
respectively, with h
m. We wish to study this clock at times be
tween 12:00 and 12:30. Let 0, 4), and L be as in Figure 32 and note
that & increases at a constant rate. By the Law of Cosines,
L = L(0) = (h
2 -b m
2
2hm cos 0)1/2, and so
Figure3l
46. Determine 0 so that the area of the symmetric cross
shown in Figure 31 is maximized. Then find this maximum area.
Figure 30
(b) minimize the area of triangle B;
(c) minimize the length z.
56. Use the information in Problems 54 and 55 to write an ex
pression for the total monthly profit P(n), n
100.
52. Brass is produced in long rolls of a thin sheet. To monitor
the quality, inspectors select at random a piece of the sheet, meas
ure its area, and count the number of surface imperfections on
that piece. The area varies from piece to piece. The following
table gives data on the area (in square feet) of the selected piece
and the number of surface imperfections found on that piece.
3
12
9
5
8
1.0
4.0
3.6
1.5
3.0
Piece
1
2
3
4
5
=
(number of hours to produce a lot of size x) + 5
25
38
7
38
52
29
Total Labor
Hours y
10
11
16
8
Lot Size x
=
—
(5 + bx)]
2
(b) Use this formula to estimate the slope b.
(c) Use your least-squares line to predict the total number of
labor hours to produce a lot consisting of 15 bookcases.
s
(a) From the description of the problem, the least-squares line
should have 5 as its y-intercept. Find a formula for the value
of the slope b that minimizes the sum of squares
1
2
3
4
Order
Some data on XYZ’s bookcases are given in the following table.
y
53. Suppose that every customer order taken by the XYZ
Company requires exacty 5 hours of labor for handling the
paperwork; this length of time is fixed and does not vary from lot
to lot. The total number of hours y required to manufacture and
sell a lot of size x would then be
E
(a) Make a scatter plot with area on the horizontal axis and
number of surface imperfections on the vertical axis.
(b) Does it look like a line through the origin would be a good
model for these data? Explain.
(c) Find the equation of the least-squares line through the
origin.
(d) Use the result of part (c) to predict how many surface im
perfections there would be on a sheet with area 2.0 square
feet.
Number of
Surface Imperfections
Area in
Square Feet
I
=
1000 + 33x
—
2 + x
9x
3
=
20 + 4x
—
=
(182
—
2
/
1
x/36)
=
800/(x + 3)
—
3
64. A riverboat company offers a fraternal organization a
Fourth of July excursion with the understanding that there will be
at least 400 passengers. The price of each ticket will be $12.00, and
the company agrees to discount the price by $0.20 for each 10
passengers in excess of 400. Write an expression for the price
function p(x) and find the number x
1 of passengers that makes
the total revenue a maximum.
find the number of units x
1 that makes the total revenue a max
imum and state the maximum possible revenue. What is the
marginal revenue when the optimum number of units, x
, is
1
sold?
p(x)
63. For the price function given by
find the number of units x
1 that makes the total revenue a maxi
mum and state the maximum possible revenue. What is the mar
ginal revenue when the optimum number of units, x
, is sold?
1
p(x)
62. For the price function defined by
where x
0 is the number of units.
(a) Find the total revenue function and the marginal revenue
function.
(b) On what interval is the total revenue increasing?
(c) For what number x is the marginal revenue a maximum?
p(x)
61. A price function,p, is defined by
Find (a) the level of production at which the marginal cost is a
minimum, and (b) the minimum marginal cost.
C(x)
60. The total cost of producing and selling lOOx units of a par
ticular commodity per week is
59. The total cost of producing and selling n units of a certain
commodity per week is C(n) = 1000 + n
/1200. Find the aver
2
age cost, C(n)/n, of each unit and the marginal cost at a produc
tion level of 800 units per week.
—
E1 58. The total cost of producing and selling x units of Xbars per
month is C(x) = 100 + 3.002x
. If the production
2
0.0001x
level is 1600 units per month, find the average cost, C(x)/x, of
each unit and the marginal cost.
57. Sketch the graph of P(n) of Problem 56, and from it esti
mate the value of n that maximizes P. Find this n exactly by the
methods of calculus.
55. The manufacturer of Zbars estimates that 100 units per
month can be sold if the unit price is $250 and that sales will in
crease by 10 units for each $5 decrease in price. Write an expres
sion for the price p(n) and the revenue R(n) if n units are sold in
one month, n
100
51. One end of a 27-foot ladder rests on the ground and the
other end rests on the top of an 8-foot wall. As the bottom end is
pushed along the ground toward the wall, the top end extends
beyond the wall. Find the maximum horizontal overhang of the
top end.
1
54. The fixed monthly cost of operating a plant that makes
Zbars is $7000, while the cost of manufacturing each unit is $100.
Write an expression for C(x), the total cost of making x Zbars in
a month.
177
50. An advertising flyer is to contain 50 square inches of
printed matter, with 2-inch margins at the top and bottom and
iinch margins on each side. What dimensions for the flyer would
use the least paper?
Section 3.4 Practical Problems
Chapter 3 Applications of the Derivative
—
3.5
Graphing Functions
Using Calculus
a+b-l-c
—
Answers to Concepts Review: 1. 0 <x <
2. 2.x -+ 200/x 3. y bx
1 4. marginal revenue; marginal cost
Hint: Consider a and c to be fixed and define F(b) =
/27b. Show that F has a minimum at b
3
(a + b + c)
. Then use
2
(a + c)/2 and that this minimum is [(a + c)/2J
the result from (b).
70. Show that of all three-dimensional boxes with a given
surface area, the cube has the greatest volume. Hint: The surface
area is S = 2(1w -1- lh + hw) and the volume is V = lwh. Let
a
1w, b = Ih, and c
11w. Use the previous problem to show
) 1/3
2
that (V
S/6. When does equality hold?
(abc)’13
EXPL 69. The arithmetic mean of the numbers a and b is
(a -1- b)/2, and the geometric mean of two positive numbers a
and b is \/ib. Suppose that a > 0 and b > 0.
(a) Show that V
(a + b)/2 holds by squaring both sides
and simplifying.
(b) Use calculus to show that Vi
(a + b)/2. Hint: Consider
a to be fixed. Square both sides of the inequality and divide
/4b.
2
through by b. Define the function F(b) = (a + b)
Show that F has its minimum at a.
(c) The geometric mean of three positive numbers a, b, and c is
. Show that the analogous inequality holds:
3
(abc)’
=
3
5 —20x
3x
f(x)=
4
15x
32
—
60x2
=
x
15x
(
2
2)(x -I- 2)
32
—
SOLUTION Since f(—x) = —f(x), f is an odd function, and therefore its graph
0, we find the x-intercepts
is symmetric with respect to the origin. Setting f(x)
±2.6. We can go this far without calculus.
to be C and hV’ö7
When we differentiate f, we obtain
• EXAMPLE i Sketch the graph of f(x)
graph by hand; one of degree 50 could be next to impossible. If the degree is of
modest size, such as 3 to 6, we can use the tools of calculus to great advantage.
Polynomial Functions A polynomial function of degree 1 or 2 is easy to
Our treatment of graphing in Section 0.4 was elementary. We proposed plotting
enough points so that the essential features of the graph were clear. We mentioned
that symmetries of the graph could reduce the effort involved. We suggested that
one should be alert to possible asymptotes. But if the equation to be graphed is
complicated or if we want a very accurate graph, the techniques of that section are
inadequate.
Calculus provides a powerful tool for analyzing the fine structure of a graph,
especially in identifying those points where the character of the graph changes.
We can locate local maximum points, local minimum points, and inflection points;
we can determine precisely where the graph is increasing or where it is concave up.
Inclusion of all these ideas in our graphing procedure is the program for this
section.
—
that it can produce up to 450 units each month, its monthly cost
2 for
0.01x
function takes the form C(x) = 800 -4- 3x
300 < r 450. Find the production level that maximizes
monthly profit and evaluate this profit. Sketch the graph of the
x < 450.
monthly profit function P(x) on 0
E1 68. If the company of Problem 67 expands its facilities so
—
produced each month. Its total monthly cost is
. At peak production, it can make
2
0.01.t
C(x) = 200 -1- 4x
300 units. What is its maximum monthly profit and what level of
production gives this profit?
The ZEE Company makes zingos, which it markets at
10 — 0.00L dollars, where x is the number
a price of p(x)
E1 67.
costs $3000.
66. Repeat Problem 65, assuming that the additional machine
—
65. The XYZ Company manufactures wicker chairs. With
its present machines it has a maximum yearly output of 500 units.
If it makes x chairs, it can set a price of p(x) 200 0.15x
dollars each and will have a total yearly cost of
2 dollars. The company has the
C(x) = 5000 + 6x 0.002x
opportunity to buy a new machine for $4000 with which the
company can make up to an additional 250 chairs per year.
The cost function for values of x between 500 and 750 is thus
. Basing your analysis on the profit
2
C(x) = 9000 + 6x — 0.002i
for the next year, answer the following questions.
(a) Should the company purchase the additional machine?
(b) What should be the level of production?
178
Figure 2
f”-
Figure 1
9
+
0
I
0—
0
I
0
2
—9—9
+
+
179
3
60x
32
—
120x
=
15x(x
—
8
+
—3
f(x)
_‘I/
=
5 — 20x)/32
(3x
—2
Local max
(—2.2)
f”<O
concave—
down
—I
—2
f•’<O
concave
down
f”>o
—l
f’.<O
decreasing
concave
up
I
0
4
J
f’>O
(2. —2)
Local mm
3
increasing
f,,>0
—concav
up
I
2
4
,P
X
B
21 Sketch the graph of f(x)
=
2
x
2x + 4
x—2
—
lim
x—.2
—
x—2x+4
2
x
—00
and
tim
x—2
—
2x+4
x
—
2
2
x
no
SOLUTION This function is neither even nor odd, so we do not have any of
the usual symmetries. There are no x-intercepts, since the solutions to
— 2x + 4 = 0 are not real numbers. The y-intercept is —2. We anticipate a
2. In fact,
vertical asymptote at x
• EXAMPLE
Rational Functions A rational function, being the quotient of two polynomi
al functions, is considerably more complicated to graph than a polynomial. In
particular, we can expect dramatic behavior near where the denominator would be
zero.
Figure 3
4
—2
f’<9
decreasing
1>9
increasing
._.
(1.4, -1.2).
Much of this information is collected at the top of Figure 3, which we use to
sketch the graph directly below it.
(\/, -7\//8)
By studying the sign of f”(x) (Figure 2), we deduce that f is concave upward on
(—v’,o) and (v’, no) and concave downward on (—no, _\/) and (o, \/).
Thus, there are three points of inflection: (—v’, 7\//8)
(—1.4,1.2), (0, 0), and
f”(x)
Thus, the critical points are —2, 0, and 2; we quickly discover (Figure 1) that
f’(x) > 0 on (—no, —2) and (2, no) and that f’(x) <0 on (—2,0) and (0, 2).
These facts tell us where f is increasing and where it is decreasing; they also con
firm that f(—2) = 2 is a local maximum value and that f(2) = —2 is a local mini
mum value.
Differentiating again, we get
Section 3.5 Graphing Functions Using Calculus