MA119-A Applied Calculus for Business
2006 Fall
Homework 3 Solutions
Due 9/22/2006 10:30AM
p
x2 +2x. Find g (a + h), g ( a), g ( a), a+g (a),
2.1 #6 Let g be the function de…ned by g (x) =
1
and g(a)
.
[Solution]
We have
g (a + h) =
(a + h)2 + 2 (a + h) =
g ( a) =
p
g a =
a2
2ah + h2 + 2a + 2h,
( a)2 + 2 ( a) = a2 2a,
p 2
p
p
a + 2 a = a 2 a,
(a)2 + 2 (a) =
a + g (a) = a +
a2 + 3a,
and if g (a) 6= 0 ( a2 + 2a 6= 0 , a 6= 0 or 2), then
1
=
g (a)
a2
1
.
+ 2a
2.1 #14 Let f be the function de…ned by
f (x) =
2+
1
1 x
p
1 + x if x 1
.
if x > 1
Find f (0), f (1), and f (2).
[Solution]
p
Since 0 1, we have f (0) = 2 + p1 + 0 = 3. p
Since 1 1, we have f (1) = 2 + 1 + 1 = 2 + 2.
Since 2 > 1, we have f (2) = 1 1 2 = 1.
2.1 #34 Find the domain of
f (x) =
p
x 1
.
(x + 2) (x 3)
[Solution]
For the numerator, we have to have x 1 0 for the square root.
For the denominator, we cannot have 0 as a denominator. So, x 6= 2 or 3.
Thus, the domain is x
1, except 3. By a mathematical notation, the domain is
fx 2 R j x 1 and x 6= 3g.
1
2
2.1 #60 Growth of a Cancerous Tumor The surface area of a spherical cancerous tumor is
given by the function
S (r) = 4 r2
where r is the radius of the tumor in centimeters. After extensive chemotherapy treatment, the surface area of the tumor is reduced by 75%. What is the radius of the tumor
after treatment?
[Solution]
Let r0 be the radius of the tumor before treatment and r1 be the radius of the tumor
before treatment. Since, after extensive chemotherapy treatment, the surface area of the
4 r2
S(r1 )
tumor is reduced by 75%, we have S(r
= 1 75% = 0:25. This implies that 4 r12 = 14 ,
0)
0
p2
r0
r0
that is, r1 = 2 = 2 . So the radius of the tumor after treatment reduced by half.
2.1 #70 Cost of Renting a Truck Ace Truck leases its 10-ft truck at $30/day and $:45/mi,
whereas Acme Truck leases a similar truck at $25/day and $:50/mi.
(a) Find the daily cost of leasing from each company as a function of the number of
miles driven.
(b) Sketch the graphs of the two functions on the same set of axes.
(c) Which company should a customer rent a truck from for 1 day if she plans to drive
at most 70 mi and wishes to minimize her cost?
[Solution]
(a) Let m be the number of miles driven a day. The daily cost of the Ace Truck is
CAce (m) = 30 + 0:45 m.
The daily cost of the Acme Truck is CAcme (m) = 25 + 0:5 m.
(b) The pink (magenta) color is the function CAce (m) for the Ace Truck and the blue
color is the function CAcme (m) for the Acme Truck.
y 60
50
Ace Truck
40
Acme Truck
30
20
10
0
0
10
20
30
40
50
60
70
80
x
(c) According to the graph, she needs to choose Acme Truck to minimize her cost.
2.1 #76 Worker E¢ ciency An e¢ ciency study conducted for Elektra Electronics showed that
the number of "Space Commander" walkie-talkies assembled by the average worker t hr
after starting work at 8:00 a.m. is given by
N (t) =
t3 + 6t2 + 15t
(0
t
4) .
3
How many walkie-talkies can an average worker be expected to assemble between 8:00
and 9:00 a.m.? Between 9:00 and 10:00 a.m.?
[Solution]
Since out t is counting after starting work at 8:00 a.m., 9:00 a.m. can be treated as
t = 1 and 10:00 a.m. can be treated as t = 2. Thus, N (1) = 13 + 6 12 + 15 1 = 20 and
N (2) = 23 + 6 22 + 15 2 = 46. Therefore, an average worker is expected to assemble 20
walkie-talkies between 8:00 and 9:00 a.m. and 46 walkie-talkies between 8:00 and 10:00
a.m. Hence, he/she is expected to assemble 46 20 = 26 walkie-talkies between 9:00 and
10:00 a.m.
2.1 #80 Postal Regulations In 2002 the postage for …rst-class mail was raised to 37c/ for the
…rst ounce or fraction thereof and 23c/ for each additional ounce or fraction thereof. Any
parcel not exceeding 12oz may be sent by …rst-class mail. Letting x denote the weight of
a parcel in ounces and f (x) the postage in cents, complete the following description of
the "postage function" f :
8
37 if 0 < x 1
>
>
< 60 if 1 < x 2
f (x) =
..
>
>
: .
? if 11 < x 12
(a) What is the domain of f ?
(b) Sketch the graph of f .
[Solution]
We can complete the function f as follow:
f (x) =
8
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
:
37
60
83
106
129
152
175
198
221
244
267
290
if
if
if
if
if
if
if
if
if
if
if
if
0<x
1<x
2<x
3<x
4<x
5<x
6<x
7<x
8<x
9<x
10 < x
11 < x
1
2
3
4
5
6
.
7
8
9
10
11
12
(a) The x in f can be any real number between 0 and 12, but not 0 itself. So, the domain
is 0 < x 12.
(b) The graph is
4
y
300
200
100
0
0
1
2
3
4
5
6
7
8
9
10
11
12
x
(Notice that each line segment does not contain the most left point.)
2.1 #82 Rising Median Age Increased longevity and the aging of the baby boom generation–
those born between 1946 and 1965–are the primary reasons for a rising median age. The
median age (in years) of the U.S. population from 1900 through 2000 is approximated by
the function
8
if 0 x
< 1:3t + 22:9
2
0:7t + 7:2t + 11:5 if 3 < x
f (x) =
: 2:6t + 9:4
if 7 < x
3
7
10
where t is measured in decades, with t = 0 corresponding to 1900.
(a) What was the median age of the U.S. population at the beginning of 1900? At the
beginning of 1950? At the beginning of 1900?
(b) Sketch the graph of f .
[Solution]
(a) At the beginning of 1900, we have t = 0. Thus, the median age is f (0) = 1:3 0 +
22:9 = 22:9.
At the beginning of 1950, we have t = 5. Thus, the median age is f (5) = 0:7 (5)2 +
7:2 (5) + 11:5 = 30.
At the beginning of 1990, we have t = 9. Thus, the median age is f (9) = 2:6 9 +
9:4 = 32:8.
(b) The graph is
5
y
34
32
30
28
26
24
0
1
2
3
4
5
6
7
8
9
10
x
p
2.2 #20 Find the function f + g, f g, f g, and f =g where f (x) = x 1 and g (x) = x3 + 1.
[Solution]
We have
p
(f + g) (x) = f (x) + g (x) = x 1 + x3 + 1,
p
(f g) (x) = f (x) g (x) = x 1 x3 1,
p
(f g) (x) = f (x) g (x) = x 1 x3 + 1 ,
and if if g (x) 6= 0 (x3 + 1 6= 0 , x 6=
1),
p
f (x)
x 1
= 3
.
(f =g) (x) =
g (x)
x +1
2.2 #34 Evaluate h (2), where h = g f and f (x) = x 1 1 and g (x) = x2 + 1.
[Solution]
Note that h (2) = g f (2) = g (f (2)). We have f (2) = 2 1 1 = 1 and g (f (2)) = g (1) =
12 + 1 = 2.
[Alternative Solution]
Since
1
h (x) = g f (x) = g (f (x)) = g
=
2
1
x
1
+1=
2
=
we have h (2) =
x2
(2)2 2(2)+2
(2)2 2(2)+1
1
x
+ 2
2x + 1 x
= 2.
x
1
(x
2
1
+1=
1)
2x + 1
x2
= 2
2x + 1
x
x2
1
+1
2x + 1
2x + 2
,
2x + 1
6
2.2 #40 Find functions f and g such that h = g f where h (x) = px12 4 . (Note: The answer is
not unique.)
[Solution] p
p
Let f (x) = x2 4 and g (x) = x1 . Then g f (x) = g (f (x)) = g x2 4 =
p 1
= h (x).
x2 4
Or, let f (x) = x2 4 and g (x) = p1x . Then g f (x) = g (f (x)) = g (x2 4) =
p 1
= h (x).
x2 4
2.2 #48 Find the simplify
f (a + h)
h
for f (x) = 2x2
[Solution]
We have
f (a + h)
h
f (a)
(h 6= 0)
x + 1.
f (a)
2 (a + h)2
(a + h) + 1
[2a2 a + 1]
h
2
2
2a + 4ah + 2h
a h + 1 2a2 + a 1
=
h
4ah + 2h2 h
=
h
= 4a + 2h 1.
=
2.2 #60 Spam Messages The total number of email messages per day (in billions) between 2003
and 2007 is forecast to be
f (x) = 1:54t2 + 7:1t + 31:4
(0
t
4)
where t is measured in years, with t = 0 corresponding to 2003. Over the same period,
the total number of spam messages per day (in billions) is forecast to be
g (x) = 1:21t2 + 6t + 14:5
(0
t
4)
(a) Find the rule for the function D = f
g. Compute D (4) and explain what is
measures.
(b) Find the rule for the function P = f =g. Compute P (4) and explain what is measures.
[Solution]
(a) D (x) = (f g) (x) = f (x) g (x) = [1:54t2 + 7:1t + 31:4 ] [1:21t2 + 6t + 14:5] =
0:33t2 + 1:1t + 16:9. So, D (4) = 0:33 (4)2 + 1:1 (4) + 16:9 = 26:58. Since f is the
total number of email messages per day and g is the total number of spam messages
per day, we know that D = f g is the number of non-spam messages per day (in
billions).
2
2 +7:1t+31:4
+7:1(4)+31:4
(x)
= 1:54t
. So, P (4) = 1:54(4)
= 0:22986.
(b) P (x) = (f =g) (x) = fg(x)
1:21t2 +6t+14:5
1:21(4)2 +6(4)14:5
Since f is the total number of email messages per day and g is the total number of
spam messages per day, we know that P = f =g is the ratio of the number of email
messages to a spam message per day.
7
2.2 #66 E¤ect of Mortgage Rates on Housing Starts A study prepared for the National
Association of Realtors estimated that the number of housing starts per year over the
next 5 yr will be
7
N (r) =
1 + 0:02r2
million units, where r (percent) is the mortgage rate. Suppose the mortgage rate over
the next t mo is
10t + 150
r (t) =
(0 t 24)
t + 10
percent/year.
(a) Find an expression for the number of housing starts per year as a function of t, t mo
from now.
(b) Using the result from part (a), determine the number of housing starts at present,
12 mo from now, and 18 mo from now.
[Solution]
(a) Since t is t mo from now, we know that at the time t, the mortgage rate is r (t) =
10t+150
. So, at this time t, the number of housing starts per year over the next 5 yr
t+10
7
is N (r (t)) = N 10t+150
=
2.
t+10
1+0:02( 10t+150
t+10 )
7
(b) For 12 mo from now, the number of housing starts is N (r (12)) =
=
10(12)+150 2
1+0:02( (12)+10 )
1:7446 (million units). For 18 mo from now, the number of housing starts is N (r (18)) =
7
= 1:8528 (million units).
10(18)+150 2
1+0:02( (18)+10 )
2.3 #16 Find the constants m and b in the linear function f (x) = mx + b so that f (2) = 4 and
the straight line represented by f has slope 1.
[Solution]
Since the straight line is represented by f and f (2) = 4, we know that the line passes
through (2; 4). By the point-slope form, the line equation is y 4 = ( 1) (x 2). This
is y = x + 6. So, we have m = 1 and b = 6.
2.3 #24 Spending on Medical Devices The U.S. market size for medical devices (in billions of
dollars) in year t, from 1999 (t = 0) through 2005 (t = 6) is approximated by the function
S (t) = 0:288t2 + 3:03t + 45:9
(0
t
6) .
What was the U.S. market size for medical devices in 1999? In 2005?
[Solution]
The U.S. market size for medical devices in 1999 is S (0) = 0:288 (0)2 +3:03 (0)+45:9 =
45:9 (in billions of dollars). And, the U.S. market size for medical devices in 2005 is
S (6) = 0:288 (6)2 + 3:03 (6) + 45:9 = 74:448 (in billions of dollars)
2.3 #38 Cricket Chirping and Temperature Entomologists have discovered that a linear relationship exists between the number of chirps of crickets of a certain species and the
air temperature. When the temperature is 70 F, the crickets chirp at the rate of 120
times/minute, and when the temperature is 80 F, the crickets chirp at the rate of 160
times/minute.
8
(a) Find an equation giving the relationship between the air temperature T and the
number of chirps/minute, N , of the crickets.
(b) Find N as a function of T and use this formula to determine the rate at which the
crickets chirp when the temperature is 102 F.
[Solution]
(a) We know that a linear relationship (which is a line) exists between the number of
chirps of crickets of a certain species and the air temperature. Also, we have two
points in this line which are (T; N ) = (70; 120) and (80; 160). We can write down a
line equation by
N
120 =
160
80
120
(T
70
70) .
So, the relationship between T and N is N = 4T 160.
(b) By (a), as a function of T , we have N (T ) = 4T 160. So, when the temperature is
102 F, the rate is N (102) = 4 (102) 160 = 248.
2.3 #44 Price of Ivory According to the World Wildlife Fund, a group in the forefront of the
…ght against illegal ivory trade, the price of ivory (in dollars/kilo) complied from a variety
legal and black market source is approximated by the function
f (x) =
8:37t + 7:44 if 0 t
2:84t + 51:68 if 8 < t
8
30
where t is measured in years, with t = 0 corresponding to the beginning of 1970.
(a) Sketch the graph of the function f .
(b) What was the price of ivory at the beginning of 1970? At the beginning of 1990?
[Solution]
(a) The graph is
y
120
100
80
60
40
20
0
0
2
4
6
8
10
12
14
16
18
20
22
24
26
28
30
x
(b) The price of ivory at the beginning of 1970 is f (0) = 8:37 (0) + 7:44 = 7:44. The
price of ivory at the beginning of 1990 is f (20) = 8:37 (20) + 7:44 = 174:84.
9
2.3 #70 Market Equilibrium The weekly demand and supply function for Sportsman 5
tents are given by
7
p =
0:1x2 x + 40
p = 0:1x2 + 2x + 20
respectively, where p is measured in dollars and x is measured in units of a hundred. Find
the equilibrium quantity and price.
[Solution]
The equilibrium means the price of supply equals the price of demand. So, we set
0:1x2
x + 40 = 0:1x2 + 2x + 20.
This is
3
2
0:2x
p 2 + 3x
3
20 = 0.
4 0:2 ( 20)
2 0:2
2
By solving it, we have x =
= 5 or 20. So the equilibrium quantity
is 5. And, the equilibrium price is 0:1 (5) + 2 (5) + 20 = 32:5.
2.3 #74 Packaging By cutting away identical squares from each corner of a rectangular piece of
card board and folding up the resulting ‡aps, an open box may be made. If the card
board is 15 in. long and 8 in. wide and the square cutaways have dimensions of x in. by
x in., …nd a function giving the volume of the resulting box.
[Solution]
The length of the box is 15 2x. The wide of the box is 8 2x. And, the height of
the box is x. Thus, the volume as a function of x is
V (x) = (15
2x) (8
2x) x.
Note that the length, the wide and the height are all positive. So, we have 15 2x > 0,
8 2x > 0 and x > 0. These imply that x < 15
, x < 4 and x > 0. Thus, the domain of
2
our function V (x) is 0 < x < 4.
2.3 #78 Book Design A book designer has decided that the page of a book should have 1-in.
margins at the top and bottom and 12 -in. margins on the sides. She further stipulated
that each page should have an area 50 in.2 . Find a function in the variable x, giving the
area of the printed page. What is domain of the function?
[Solution]
From the …gure in the textbook page 91, we have that xy is the area of each page, which
is 50. Thus, y = 50
. The length of the x-side of the printed page is x 2 12 = x 1.
x
And, the length of the y-side of the printed page is y 2 1 = 50
2. Therefore, a
x
function giving the area of the printed page is
50
f (x) = (x 1)
2 .
x
We must have positive length of both sides. So, we have x 1 > 0 and 50
2 > 0. These
x
imply that x > 1 and x < 25. Therefore, the domain of this function is 1 < x < 25.