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Unformatted text preview: 210 CHAPTER 2 Functions We can now express the areas of the top, bottom, and sides in terms of r only. "in Words _ . in Algebra
Radius of can r
Height of can 1003
WT Area of top and bottom 247:2  . 1000
Area of Sldes {272741) 2m 2 WV E Set up the Modei The model is the function S that gives the surface area of the can as a function of radius r.
surface area ”—‘” area of top and bottom + area of sides
I S{r) 2 27rr2 + 27:7(1003)
I arr
. E SO“) = 2,71%: vi~ 20:)0
l l
L .
' mmmmehm___.._.__.;_.m. ___._./ g 5
0 19 Use the Model
Figure 9 We use the mode} to find the minimum surface area of the can, We graph Sin
S = 2W2 + 2000 Figure 9 and zoom in on the minimum point to find that the minimum value of
r S is about 554 cm2 and occurs when the radius is about 5 .4 cm.
Exercises 1—18 m In these exercises you are asked to find a function that
models a real~1ife situation. Use the guideiines for modeling
described in the text to help you. 1. Area A rectangular building iot is three times as long as it is wide. Find a function that models its area A in toms of its
width w. 2. Area A poster is 10 inches ionger than it is wide. Find a
function that models its area A in terms of its width w. Volume A rectangular box has a square base. Its height is
half the width of the base. Find a function that models its
voinme V in terms of its width to. Volume The height of a cylinder is four times its radius.
Find a function that models the volume V of the cylinder in
terms of its radius r. 5. Area A rectangle has a perimeter of 20 it. Find a function
that models its area A in terms of the length x of one of its
sides. 6. 10. 11. 12. . Area Find a function that models the area A of an equila . Area Find a function that models the surface area S of a Perimeter A rectangle has an area of 16 m2. Find a fun
tion that models its perimeter P in terms of the length x of
one of its sides. eral triangle in terms of the length x of one of its sides. cube in terms of its volume V. Radius Find a function that models the radius r of a circ
in terms of its area A. Area Find a function that models the area A of a circlci
terms of its cirCumference C. Area A rectangular box with a volume of 60 ft3 has a
square base. Find a function that models its surface area 5
terms of the iength x of one side of its base. Length A woman 5 ft tail is standing near a street lamp
that is 12 ft tall, as shown in the ﬁgure. Find a function til r only. models the length L of her shadow in terms of her distance d from the base of the lamp.  3. Distance Two ships ieave port at the same time. One sails
south at £5 mi/h and the other sails east at 20 rnifh. Find a 8 function that models the distance D between the ships in
terms of the time t (in hours) eiapsed since their departure. h S in vaiue of 4. i’roduct The sum of two positive numbers is 60. Find a
function that modeis their product P in terms of x, one of the
numbers. 5. Area An isosceles triangle has a perimeter of 8 cm. Find a function that modeis its area A in terms of the length of its
base b.  5 Pﬁrirneter A right triangle has one leg twice as tong as
the other. Find a function that models its perimeter P in
 terms of the length x of the shorter leg. ' Area A rectangle is inscribed in a semicircle of radius 10,
.' as show in the ﬁgure. Find a function that models the area
A of the rectangle in terms of its height h. A of a circle 60 ft3 has a
surface area r a street 1 SECTTON 2.6 Modeling with Functions 21‘! 19—36 E In these problems you are asked to ﬁnd a function that
models 'a realwiife situation, and then use the model to answer
questions about the situation. Use the guidelines on page 205 to
help you. 19. Maximizing a Product Consider the foilowing problem:
Find two numbers whose sum is 19 and whose product is as
large as possible. (3) Experiment with the problem by making a table like the
one below, showing the product of different pairs of
numbers that add up to 19. Based on the evidence in
your table, estimate the answer to the problem. First number Secondnumber Product 03) Find a function that modeis the product in terms of one
of the two numbers. (c) Use your model to solve the problem, and compare with
your answer to part (a). 20. Minimizing a Sum Find two positive numbers whose
sum is 100 and the sum of whose squares is a minimum. 21. Maximizing a Product Find two numbers whose sum is
""24 and whose product is a maximum. 22 Maximizing Area Among all rectangies that have a
perimeter of 20 ft, find the dimensions of the one with the Iargest area. 23. Fencing a Field Consider the foilowing problem: A
farmer has 2400 ft of fencing and wants to fence off a
rectangular field that borders a straight river. He does not
need a fence aiong the river (see the ﬁgure). What are the
dimensions of the ﬁeld of largest area that he can fence? (3) Experiment with the problem by drawing several dia~
grams iilustrating the situation. Caiculate the area of
each conﬁguration, and use your results to estimate the
dimensions of the largest possible ﬁeld. (b) Find a function that models the area of the ﬁeld in terms
of one of its sides. to) Use your model to soive the problem, and compare with
your answer to part (a). 212. 24. 25. m
3% 26. CHAPTER 2 Functions Dividing a Pen A rancher with 750 ft of fencing wants to enclose a rectangular area and then divide it into four pens with fencing parallel to one side of the rectangle (see the ﬁgure). (a) Find a function that models the total area of the four
pens. (is) Find the largest possible total area of the four pens. Fencing 3 Garden Plot A property owner wants to fence
a garden plot adjacent to a road, as shown in the ﬁgure. The ‘ fencing next to the road must be sturdicr and costs $5 per
foot, but the other fencing costs,‘ just $3 per foot. The garden
is to have an area of 1200 ftz. (3) Find a function that models the cost of fencing the
garden. ' (1)) Find the garden dimensions that minimize the cost of
fencing. (c) If the owner has at most $600 to spend on fencing, ﬁnd
the range of lengths he can fence along the road. Maximizing Area A wire 10 cm long is cut into two
pieces, one of length x and the other of length 10 — x,
as shown in the ﬁgure. Each piece is bent into the shape
of a square. (3) Find a function that models the total area enclosed by
the two squares. (1)) Find the value of x that minimizes the total area of the
two squares. 27. Stadium Revenue A baseball team plays in a stadium that holds 55,000 spectators. With the ticket price at $10,
the average attendance at recent games has been 27,000.
A market survey indicates that for every dollar the ticket
price is iowered, attendance increases by 3000. (3) Find a function that models the revenue in terms of
ticket price. (b) What ticket price is so high that no revenue is
generated? (c) Find the price that maximizes revenue from ticket
sales. Maximizing Profit A community birchwatching society
makes and sells simple bird feeders to raise money for its
conservation activities. The materials for each feeder cost
$6, and they sell an average of 20 per week at a price of
$10 each. They have been considering raising the price.
so they conduct a survey and ﬁnd that for every dollar
increase they lose 2 sales per week. {3) Find a function that models weekly profit in terms of
price per feeder. (it) What price should the society charge for each
feeder to maximize proﬁts? What is the maximum
proﬁt? . Light from a Window A Norman window has the shape of a rectangie surmounted by a semicircle, as shown
in the ﬁgure. A Norman window with perimeter 30 ft is to
he constructed. (a) Find a function that models the area of the
window. (in) Find the dimensions of the window that admits the
greatest amount of light. 38. Volume of a Box A hex with an open top is to be constructed from a rectanguiar piece of cardboard with
dimensions 12 in. by 20 in. by cutting out equal squares
of side it at each corner and then foiding up the sides
(see the figure). (a) Find a function that models the volume ofthe box._f (1,} Find the values of x for which the volume is greater 1. a Stadium . 3
ice at $13, I than 200 m .
3121000. I (c) Find the largest volume that such a box can have. ‘ the ticket terms of te is
5?
masher
. _ . sag 35.
g 31' Area of a Box An oeen hex With a square base is to have
may for it a volume of £2 fta.
ilfé'ﬁdaf cost (3) Find a functiou that models the surface area of the
ta‘price of box.
the price, _ (13) Find the box dimensions that minimize the amount of
W doilar material used.
1511 terms 0f. 32 Inscribed Rectangle Find the dimensions that give the
' largest area for the rectangle shown in the ﬁgure. Its base is
each  on the mods and its other two vertices are above the x—axis,
naximurn lying on the parabola y = 8  x2.
a
3’
I has the a
tcle, as shown
:ter 30 ft is t
to
admits the
gg 36.
Minimizing Costs A rancher wants to build a rectangular
pen with an area of 100 mg.
(a) Find a function that models the length of fencing
required.
lb} Find the pen dimensions that require the minimum
amount of fencing.
Minimizing 'i'ime A man stands at a point A on the
bank of a straight river, 2 mi wide. To reach point B,
lmi downstream on the opposite bank, he ﬁrst rows
I i) is to be his boat to point P on the opposite bani: and then wants
lboard with the remaining distance x to B, as shown in the figure.
.qual square He can row at a speed of 2 mi/h and walk at a speed of
the sides '5 mu’h. a) Find a function that models the time needed for
z of the box ' SECTION 2.6 Modeling with Functions 213 (1)) Where should he iand so that he reaches B as soon as
possible? Bird Flight A bird is reteased from pointA on an island,
5 mi from the nearest point B on a straight shoreline. The
bird ﬂies to a point C on the shoreline, and then flies along
the shoreline to its nesting area I) (see the ﬁgure}. Suppose
the bird requires 10 itcalfrni of energy to ﬂy over land and
E4 kcal/mi to fly over water (see Example 9 in Section 1.6). (a) Find a function that models the energy expenditure of
the bird. (b) If the bird instinctively chooses a path that minimizes
its energy expenditure, to what goint does it fly? Area of a Kite A kite frame is to be made from six pieces
of wood. The four pieces that form its border have been cut
to the lengths indicated in the ﬁgure. Let x be as shown in
the ﬁgure. (a) Show that the area of the kite is given hy the function A(x) = x( V25 — x2 + 144  x2)
(1)) How long shouid each of the two crosspieces be to
maximize the area of the kite? ...
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 Spring '06
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