MATH 1342 - chapter 1
Name and Section Number (10 points)
Each question below is 10 points unless otherwise noted.
Find the slope of the line.
1)
Solve the problem.
8) The change in a certain engineer's salary over
time can be approximated by the linear
equation y = 1500x + 47,500 where y represents
salary in dollars and x represents number of
years on the job. According to this equation,
after how many years on the job was the
engineer's salary $64,000?
y
10
5
-10
-5
5
10
x
9) Suppose the sales of a particular brand of
appliance satisfy the relationship
S(x) = 150x + 1700, where S(x) represents the
number of sales in year x, with x = 0
corresponding to 1982. Find the number of
sales in 1997.
-5
-10
10) The information in the chart gives the salary of
a person for the stated years. Model the data
with a linear function using the points
(1, 24,600) and (3, 26,700).
Year, x
Salary, y
1990, 0
$23,500
1991, 1
$24,600
1992, 2
$25,200
1993, 3
$26,700
1994, 4
$27,200
2)
y
10
5
-10
-5
5
10
x
-5
11) After two years on the job, an engineer's salary
was $60,000. After seven years on the job, her
salary was $66,000. Let y represent her salary
after x years on the job. Assuming that the
change in her salary over time can be
approximated by a straight line, give an
equation for this line in the form y = mx + b.
-10
3) A line parallel to 4y - 3x = -8
4) 4x - 2y = -2
Write an equation for the line. Use slope-intercept form, if
possible.
12) Through (-4, -2), m = -2.5
5) y = 9x
Find the slope of the line passing through the given pair
of points.
6) (6, 4) and (-9, 2)
13) Through (13, -5), m = -4
14) Through (3, 10), m = 0
7) (4, 2) and (8, 9)
15) Through (-10, 6) and (-10, 9)
16) Through (-6, -8) and (-3, 3)
1
17) Through (5, 10) and (-3, 10)
26) A book publisher found that the cost to
produce 1000 calculus textbooks is $26,100,
while the cost to produce 2000 calculus
textbooks is $51,400. Assume that the cost C(x)
is a linear function of x, the number of
textbooks produced. What is the marginal cost
of a calculus textbook?
Determine the equation of the line described. Put answer
in the slope-intercept form, if possible.
18) Through (6, -8), perpendicular to -3x - 8y = 46
19) Through (-7, -5), perpendicular to 2x + 7y =
-49
27) On a summer day, the surface water of a lake
is at a temperature of 21° Celsius. What is this
temperature in Fahrenheit?
20) Through (3, 3), perpendicular to -4x + 7y = -33
21) Through (-3, 8), perpendicular to x = 5
28) The outdoor temperature rises to 20°
Fahrenheit. What is this temperature in
Celsius?
Solve the problem.
22) Regrind, Inc. regrinds used typewriter platens.
The cost per platen is $1.40. The fixed cost to
run the grinding machine is $174 per day. If
the company sells the reground platens for $
3.40, how many must be reground daily to
break even?
29) The temperature of water in a certain lake on a
day in October can be determined by using the
model y = 15.2 - 0.537x where x is the number
of feet down from the surface of the lake and y
is the Celsius temperature of the water at that
depth. Based on this model, how deep in the
lake is the water 10 degrees? (Round to the
nearest foot.)
23) Midtown Delivery Service delivers packages
which cost $1.30 per package to deliver. The
fixed cost to run the delivery truck is $105 per
day. If the company charges $6.30 per package,
how many packages must be delivered daily to
break even?
30) On a summer day, the bottom water of a lake
is at a temperature of 9° Celsius. What is this
temperature in Fahrenheit?
24) Regrind, Inc. regrinds used typewriter platens.
The cost per platen is $1.90. The cost to regrind
100 platens is $400. Find the linear cost
function to regrind platens. If reground platens
sell for $9.60 each, how many must be
reground and sold to break even?
25) A toilet manufacturer has decided to come out
with a new and improved toilet. The fixed cost
for the production of this new toilet line is
$16,600 and the variable costs are $67 per
toilet. The company expects to sell the toilets
for $157. Formulate a function P(x) for the total
profit from the production and sale of x toilets.
2
31) Let the supply and demand functions for
raspberry-flavored licorice be given by
5
p = S(q) = q and
p = D(q) = 60 4
Solve the problem.
36) Let the supply and demand functions for a
certain model of electric pencil sharpener be
given by
2
p = S(q) = q and
p = D(q) = 12 3
3
q,
4
4
q,
3
where p is the price in dollars and q is the
number of batches. Graph these functions on
the same axes (graph the supply function as a
dashed line and the demand function as a
solid line). Also, find the equilibrium quantity
and the equilibrium price.
100
where p is the price in dollars and q is the
quantity of pencil sharpeners (in hundreds).
Graph these functions on the same axes (graph
the supply function as a dashed line and the
demand function as a solid line). Also, find the
equilibrium quantity and the equilibrium
price.
p
90
80
70
20
60
p
18
50
16
40
14
12
30
10
20
8
10
6
4
10 20 30 40 50 60 70 80 90 100 q
2
2
Write a cost function for the problem. Assume that the
relationship is linear.
32) Marginal cost, $40; 100 items cost $5100 to
produce
4
6
8 10 12 14 16 18 20 q
Write a cost function for the problem. Assume that the
relationship is linear.
37) Fixed cost, $160; 5 items cost $730 to produce
Provide an appropriate response.
33) Find k so that the line through (3, k) and (1, -2)
is parallel to 2x - 3y = -9. Find k so that the
line is perpendicular to 5x + 5y = 9.
Provide an appropriate response.
38) Show that the points P1 (2,4), P2(5,2), and
P3 (7,5) are the vertices of a right triangle.
34) Can an equation of a vertical line be written in
slope-intercept form? Explain.
39) The total number of reported cases of AIDS in
the United States has risen from 372 in 1981 to
100,000 in 1989 and 200,000 in 1992. Does a
linear equation fit this data? Explain.
35) If a company decides to make a new product,
there are fixed costs and variable costs
associated with this new product. Explain the
differences of the two types of costs and why
they occur. Use an example to illustrate your
point.
40) Why is the slope of a horizontal line equal to
zero? Give an example.
41) Explain what is wrong with the statement "The
line has no slope."
3
Graph the parabola. Give its vertex and axis of symmetry.
1
42) y = - x2 - 4x - 10
2
51) A projectile is thrown upward so that its
distance above the ground after t seconds is
h = -15t2 + 480t. After how many seconds does
it reach its maximum height?
43) y = 3x2 + 12x + 6
Graph the feasible region for the system of inequalities.
52) 3x - 2y ≥ -6
x- 1<0
Solve the problem.
44) John owns a hotdog stand. He has found that
his profit is represented by the equation
P(x) = -x2 + 14x + 54, where x is the number of
5
y
hotdogs. What is the most he can earn?
45) Suppose the cost of producing x items is given
by C(x) = 2800 - x3 and the revenue made on
5 x
-5
the sale of x items is R(x) = 200x - 14x2 . Find
the number of items which serves as a
break-even point.
-5
46) An advertising agency has discovered that
when the Holt Company spends x thousands
of dollars on advertising, it results in a profit
increase in thousands of dollars given by the
function
1
P(x) = - (x - 6)2 +
3
53) x + 2y ≤ 2
x+y≥0
5
y
30
5 x
-5
How much should the Holt Company spend
on advertising to maximize the profit?
47) Let C(x) = 6x + 58 be the cost to produce x units
of a product, and let R(x) = - x2 + 22x be the
-5
revenue. Find the maximum profit.
54) x - 2y ≤ 2
x+y≤0
48) Bob owns a watch repair shop. He has found
that the cost of operating his shop is given by
C(x) = 4x2 - 248x + 89, where x is the number
5
y
of watches repaired. How many watches
should he repair to produce the lowest cost?
5 x
-5
49) The number of mosquitoes M(x), in millions, in
a certain area depends on the June rainfall x, in
inches: M(x) = 4x - x2. What rainfall produces
the maximum number of mosquitoes?
-5
50) The length and width of a rectangle have a
sum of 156. What dimensions will give the
maximum area?
4
Write the system of inequalities that describes the
possible solutions to the problem.
55) A manufacturer of wooden chairs and tables
must decide in advance how many of each
item will be made in a given week. Use the
table to find the system of inequalities that
describes the manufacturer's weekly
production.
Use the indicated region of feasible solutions to find the
maximum and minimum values of the given objective
function.
58) z = 12x - 22y
y
(0, 6)
(1.2, 5)
Use x for the number of chairs and y for the
number of tables made per week. The number
of work-hours available for construction and
finishing is fixed.
Construction
Finishing
Hours Hours
Total
per
per
hours
chair table available
2
4
48
2
3
42
(5, 0)
x
59) z = 21x + 5y + 12
56) The Acme Class Ring Company designs and
sells two types of rings: the VIP and the SST.
The company can produce up to 84 rings each
day, using up to 210 total man-hours of labor.
It takes 3 man-hours to make one VIP ring and
7 man-hours to make one SST ring. Let x
represent the number of VIP rings, and let y
represent the number of SST rings. Find the
system of inequalities that describes this
company's ring production.
y
(0, 1)
(1, 0)
57) A manufacturer of wooden chairs and tables
must decide in advance how many of each
item will be made in a given week. Use the
table to find the system of inequalities that
describes the manufacturer's weekly
production.
Provide an appropriate response.
60) Does a linear program with at least three
constraints always have a closed feasible
region?
Use x for the number of chairs and y for the
number of tables made per week. The number
of work-hours available for construction and
finishing is fixed.
Construction
Finishing
x
61) Explain why the graphing method is not
satisfactory for solving a linear programming
problem with 3 variables.
62) Is it possible to have a bounded feasible region
that does not optimize an objective function?
Hours Hours
Total
per
per
hours
chair table available
2
3
36
2
2
28
5
Use graphical methods to solve the linear programming
problem.
63) Minimize
z = 0.18x + 0.12y
subject to:
65) Minimize
z = 4x + 5y
subject to:
2x - 4y ≤ 10
2x + y ≥ 15
2x + 6y ≥ 30
x≥0
4x + 2y ≥ 20
y≥ 0
x≥0
y≥ 0
y
10
y
10
-10
-10
10
10
x
10
x
x
-10
-10
66) Maximize
subject to:
64) Minimize
z = 4x + 5y
2x - 4y ≤ 10
2x + y ≥ 15
0≤x≤9
0≤y≤5
z = 6x + 8y
subject to:
2x + 4y ≥ 12
2x + y ≥ 8
x≥0
y
y≥ 0
10
y
5
10
-10
-5
5
-5
-10
10
x
-10
Provide an appropriate response.
67) A linear program is defined with constraints
10x + 3y ≥ 6, 2x + 9y ≥ 0, x ≥ 0, and y ≥ 0. Is the
feasibility region bounded, unbounded, or
empty?
-10
6
Solve the problem.
68) The Acme Class Ring Company designs and
sells two types of rings: the VIP and the SST.
They can produce up to 24 rings each day
using up to 60 total man-hours of labor. It
takes 3 man-hours to make one VIP ring and 2
man-hours to make one SST ring. How many
of each type of ring should be made daily to
maximize the company's profit, if the profit on
a VIP ring is $50 and on an SST ring is $10?
72) A summer camp wants to hire counselors and
aides to fill its staffing needs at minimum cost.
The average monthly salary of a counselor is
$2400 and the average monthly salary of an
aide is $1100. The camp can accommodate up
to 35 staff members and needs at least 20 to
run properly. They must have at least 10 aides,
and may have up to 3 aides for every 2
counselors. How many counselors and how
many aides should the camp hire to minimize
cost?
69) A certain area of forest is populated by two
species of animals, which scientists refer to as
A and B for simplicity. The forest supplies two
kinds of food, referred to as F1 and F2 . For one
73) A summer camp wants to hire counselors and
aides to fill its staffing needs at minimum cost.
The average monthly salary of a counselor is
$2400 and the average monthly salary of an
aide is $1100. The camp can accommodate up
to 45 staff members and needs at least 30 to
run properly. They must have at least 10 aides,
and may have up to 3 aides for every 2
counselors. How many counselors and how
many aides should the camp hire to minimize
cost?
year, species A requires 1.35 units of F1 and
1.2 units of F2 . Species B requires 2.2 units of
F1 and 1.8 units of F2 . The forest can normally
supply at most 830 units of F1 and 488 units of
F2 per year. What is the maximum total
number of these animals that the forest can
support?
74) Suppose that Janine desires 46 grams of
protein and 38 grams of dietary fiber daily.
One serving of kidney beans has 8 grams of
protein and 6 grams of dietary fiber. One
serving of refried pinto beans has 6 grams of
protein and 6 grams of dietary fiber. If a
serving of kidney beans costs $0.45 and a
serving of refried pinto beans costs $0.35, then
how many servings of each should Janine eat
to minimize cost and still meet her
requirements?
70) Zach is planning to invest up to $50,000 in
corporate and municipal bonds. The least he
will invest in corporate bonds is $6000 and he
does not want to invest more than $29,000 in
corporate bonds. He also does not want to
invest more than $28,174 in municipal bonds.
The interest is 8.5% on corporate bonds and
6.8% on municipal bonds. This is simple
interest for one year. What is the maximum
value of his investment after one year?
71) The Acme Class Ring Company designs and
sells two types of rings: the VIP and the SST.
They can produce up to 24 rings each day
using up to 60 total man-hours of labor. It
takes 3 man-hours to make one VIP ring and 2
man-hours to make one SST ring. How many
of each type of ring should be made daily to
maximize the company's profit, if the profit on
a VIP ring is $20 and on an SST ring is $50?
75) Suppose an animal feed to be mixed from
soybean meal and oats must contain at least
100 lb of protein, 20 lb of fat, and 12 lb of
mineral ash. Each 100-lb sack of soybean meal
costs $20 and contains 50 lb of protein, 10 lb of
fat, and 8 lb of mineral ash. Each 100-lb sack of
alfalfa costs $11 and contains 30 lb of protein, 8
lb of fat, and 3 lb of mineral ash. How many
sacks of each should be used to satisfy the
minimum requirements at minimum cost?
7
Answer Key
Testname: 1324-PT-1-3-10
1) 1
2) Undefined
3
3)
4
100
35) Fixed costs occur
only once. These
costs may be startup
costs related to the
production of the
new product.
Variable costs depend
on how much
product is made.
These costs may
consist of labor,
material, and
maintenance.
p
90
80
4) 2
5) 9
2
6)
15
70
60
50
7
7)
4
40
30
8) 11 years
9) 3950
10) y = 1050x + 23,500
11) y = 1200x + 57,600
12) y = -2.5x - 12
13) y = -4x + 47
14) y = 10
15) x = -10
11
16) y =
x + 14
3
17) y = 10
8
18) y = x - 24
3
19) y =
31)
7
39
x+
2
2
20) y = -
7
33
x+
4
4
21) y = 8
22) 87 platens
23) 21 packages
24) C(x) = 1.90x + 210
break-even = 27
25) P(x) = 90x - 16600
26) $25.30
27) 69.8°
28) -6.7°
29) 10
30) 48.2°
20
10
10 20 30 40 50
For example, a
company decided to
make oak filing
cabinets. Fixed costs
would include the
costs of purchasing
and renovating plant
space and the cost of
manufacturing
equipment. Variable
costs would include
the cost labor and the
cost of materials.
Equilibrium
quantity: 30
Equilibrium price: $
37.50
32) C(x) = 40x + 1100
2
33) - ; 0
3
34) No. In the
slope-intercept form
of the equation of a
line, x is multiplied
by slope; however,
the slope of a vertical
line is undefined.
(Explanations will
vary.)
36)
20
p
18
16
14
12
10
8
6
4
-2/3. The slope of the
line through P2 and
P3 is 3/2. Therefore,
since the product of
these slopes is -1, the
lines are
perpendicular and
constitute a right
angle in the triangle,
making the triangle
formed by these
points a right
triangle.
39) No, the data cannot
be modeled by a
linear equation
because the reported
cases are not
increasing at a
constant rate.
Assume a linear
equation, and
examine the slope of
the two line
segments. The slope
of the segment from
(0, 372) to (8, 100,000)
is 12,453.5 while the
slope of the segment
from (8, 100,000) to
(11, 200,000) is
33,333.
3.(Explanations will
vary.)
2
2
4
6
8 10 12
Equilibrium
quantity: 600
Equilibrium price: $4
37) C(x) = 114x + 160
8
38) Answers will vary.
One possibility: The
slope of the line
through P1 and P2 is
Answer Key
Testname: 1324-PT-1-3-10
40) Answers may vary.
One possibility: The
slope of a horizontal
line is equal to zero
because the y-values
do not change as the
x-values change. For
example, the points
(3, 4) and (7, 4) are
two points on a
horizontal line. The
slope of this line is
zero because m =
4-4 0
= = 0.
7-3 4
41) Answers may vary.
One possibility: It is
not specific enough.
The slope of a
horizontal line is 0,
while the slope of a
vertical line is
undefined.
42)
15
y
43)
54)
10
y
5
8
6
4
2
-5
-10 -8
-6
-4
-2
-2
-4
-5
-6
55) 2x + 4y ≤ 48
2x + 3y ≤ 42
x≥0
y≥0
56) x + y ≤ 84
3x + 7y ≤ 210
x ≥ 0, y ≥ 0
57) 2x + 3y ≤ 36
2x + 2y ≤ 28
x≥0
y≥0
58) Maximum of 60;
minimum of -132
59) No maximum;
minimum of 17
60) No
61) This problem would
have to be graphed in
three dimensions
with the feasible
region in three
dimensions. This is
difficult to do on a
two-dimensional
sheet of paper.
62) No
63) Minimum of 1.02
when x = 3 and y = 4
92
64) Minimum of
3
-8
-10
Vertex: (-2, -6); Axis:
x = -2
44) $103
45) 14 items
46) $6000
47) $6
48) 31 watches
49) 2 inches
50) 78 by 78
51) 16 seconds
52)
12
5
y
9
6
3
-15 -12 -9
-6
-3
-5
-3
-6
-9
-12
-15
y
-5
53)
Vertex: (-4, -2); Axis:
x = -4
5
y
when x =
-5
10
and y =
3
4
3
65) Minimum of 33 when
x = 7 and y = 1
-5
9
66) Maximum of 61
when x = 9 and y = 5
67) Unbounded
68) 20 VIP and 0 SST
69) 406 animals
70) $53,893
71) 0 VIP and 24 SST
72) 8 counselors and 12
aides
73) 12 counselors and 18
aides
7
74) servings of refried
3
pinto beans and 4
servings of kidney
beans
2
75) sacks of soybeans
3
and
20
sacks of
9
alfalfa