Fall 2015
Math 141:505
Exam 1
Form A
Last Name:
Exam Seat #:
First Name:
UIN:
On my honor, as an Aggie, I have neither given nor received unauthorized aid on
this academic work.
Signature:
INSTRUCTIONS
• Part 1: Multiple Choice (Problems 1-8) Each problem is worth 6 points for a total of 48 points. Answers
should be written in the boxes provided. No partial credit will be given.
• Part 2: Work Out (Problems 9-13) The number of points for each problem is indicated next to the problem for
a total of 52 points. Partial credit will be given. All steps must be written clearly and neatly to receive credit.
If you use your calculator for anything beyond an arithmetic calculation, please indicate how at the appropriate
step. Box your final answer.
“An Aggie does not lie, cheat or steal or tolerate those who do.”
c
Yeong-Chyuan
Chung, September 25, 2015
Part 1: Multiple-Choice. Each problem is worth 6 points. No partial credit will be given. Answers should
be written in the boxes provided.
1. Given the equation 2x + 3y = 5, if x decreases by 2 units, what is the corresponding change in y?
(a) Decreases by 3 units.
(b) Increases by 3 units.
(c) Decreases by 43 units.
(d) Increases by
4
3
units.
d
2. An office building was worth $1,000,000 in 2005. It is depreciated linearly over 45 years. If the book value of
the building is $780,000 at the end of 2015, what is the scrap value of the building?
(a) $0
(b) $5,000
(c) $10,000
(d) $22,000
c
3. Solve the following system of linear equations:
x − 2y + 3z = 4
2x + 3y = z + 2
x + 2y − 3z = −6
17
23
,z=
7
7
(b) x = −1, y = 5, z = 5
(c) x = 5, y = −1, z = 5
(a) x = −1, y =
(d) No solution
a
2
c
Yeong-Chyuan
Chung, September 25, 2015


1 1 1 3
4. Given the augmented matrix  0 2 1 2 , which of the following is obtained after performing the operation
2 2 1 1
R3 − 2R1 , followed by the operation 12 R2 ?


−3 −3 −1 1
2
1 1 
(a)  2
0
2
1 2


1 1 1
3
(b)  0 1 1/2 1 
0 0 −1 −5


1 1 1
3
2 
(c)  0 2 1
0 0 −1 −5


3
1 1 1
(d)  0 0 −1 −5 
0 2 1
2
b
T

5
2 5
1 1 2 3
1 c
3


5. Given the matrix equation a 3 −
=
, find a + b + c.
9b 6 3
1 1 3
3
1 4
5
3
(b) 3
11
(c)
3
7
(d)
3
(a)
b


1 0 a 0
6. Which of the following values of a and b would make the matrix  0 b 1 0  be in row-reduced form?
0 0 0 1
(a) a = 3, b = 0
(b) a = 2, b = 0
(c) a = 1, b = 0
(d) a = 1, b = 1
d
3
c
Yeong-Chyuan
Chung, September 25, 2015
7. Cindy regularly makes long-distance phone calls to three foreign cities: London, Tokyo, and Hong Kong. The
matrices A and B give the lengths (in minutes) of her calls during peak and nonpeak hours, respectively, to each
of these three cities during the month of June.
London Tokyo
80
60
A=
Hong Kong
40
London
300
B=
Tokyo
150
Hong Kong
250
The costs for the calls (in dollars per minute) for the peak and nonpeak periods in June are given, respectively,
by the matrices


0.12
London
 0.15 
Tokyo
D=
0.17
Hong Kong


0.17
London
 0.21 
Tokyo
C=
0.24
Hong Kong
Which of the following matrices represents Cindy’s total cost (in dollars) for these calls?
(a) CA + DB
(b) AC + BD
(c) CAT + DBT
(d) ACT + BDT
b
8. Which sequence of row operations will allow you to pivot the given matrix about the entry that is in the second
row and second column?


1 2
3 3
 0 −5 5 2 
0 4 −2 1
(a) − 51 R2 , then R1 − 2R2 , and R3 − 4R2
(b)
(c)
1
4 R3 , then R1 − 2R3 , and
− 15 R2 , then R3 − 4R2
R2 + 5R3
(d) None of the above
a
4
c
Yeong-Chyuan
Chung, September 25, 2015
Part 2: Work Out. The number of points for each problem is indicated next to the problem. Partial credit will
be given. All steps must be written clearly and neatly to receive credit. If you use your calculator for anything
beyond an arithmetic calculation, please indicate how at the appropriate step. Box your final answer.
9. (8pts) The supply and demand for a particular commodity are given, respectively, by the equations
x − 300p + 600 = 0
and
2x + 800p − 4000 = 0,
where p denotes the unit price in thousands of dollars, and x denotes the quantity in thousands. Find the equilibrium quantity and the equilibrium price. Round your answers to the nearest integer.
Solution:
Rewrite the equations as
1
x+2
300
1
p=−
x+5
400
p=
Then graph the two lines and use the Intersect command to find the intersection point, which is (514.3, 3.714).
The equilibrium quantity is 514,300, and the equilibrium price is $3714.
Alternatively, use the rref command to solve the system of linear equations.
1 −300 −600 rre f 1 0 3600/7
26
In this case, you will get
→
so x = 3600
7 and p = 7 , which means the
2 800 4000
0 1 26/7
26
equilibrium quantity is 3600
7 (1000) = 514, 286, and the equilibrium price is $ 7 (1000) = $3714.
10. (12pts) A stadium is trying to analyze the relationship between ticket prices and demand (the number of tickets
1
sold). The demand equation was found to be p = − 500
x + 264, where p is the price in dollars, and x is the
number of tickets sold.
(a) Find the revenue function.
(b) Find the vertex of the revenue function without graphing the function.
(c) How many tickets need to be sold to maximize revenue? What is the maximum revenue?
Solution:
1
(a) R(x) = x · p(x) = x(− 500
x + 264).
1 2
The revenue function is R(x) = − 500
x + 264x.
b
264
(b) x = − 2a
= − −2/500
= 66000. Plugging this into the revenue function yields R(66000) = 8712000. The
vertex of the revenue function is (66000, 8712000).
(c) 66,000 tickets need to be sold. The maximum revenue is $8,712,000.
5
c
Yeong-Chyuan
Chung, September 25, 2015
11. (10pts) Consider the following word problem.
“A plush toy manufacturer makes Elsa, Olaf, and Chewbacca plush toys. Each Elsa toy requires 2 yd2 of plush,
30 ft3 of stuffing, and 15 pieces of trim. Each Olaf toy requires 1.5 yd2 of plush, 20 ft3 of stuffing, and 10 pieces
of trim. Each Chewbacca toy requires 3 yd2 of plush, 50 ft3 of stuffing, and 25 pieces of trim. The manufacturer
has 3500 yd2 of plush, 50,000 ft3 of stuffing, and 25,000 pieces of trim, and they want to use up all the material.
They also want to make twice as many Olaf toys as Elsa toys. How many toys of each type should they make?”
Clearly define the appropriate variables, and set up (but do not solve) a system of linear equations that would be
used to answer the question.
Solution:
Let x be the number of Elsa toys.
Let y be the number of Olaf toys.
Let z be the number of Chewbacca toys.
2x + 1.5y + 3z = 3500
30x + 20y + 50z = 50000
15x + 10y + 25z = 25000
2x − y = 0
1 x
3 0 5
12. (10pts) Let A =
and let B =
where x is a constant.
3 2
1 2 4
(a) Which of the two products AB and BA is undefined? Explain why.
(b) Compute the product that is defined.
Solution:
(a) BA is undefined because B has 3 columns and A has 2 rows.
1(3) + x(1) 1(0) + x(2) 1(5) + x(4)
3 + x 2x 5 + 4x
(b) AB =
=
.
3(3) + 2(1) 3(0) + 2(2) 3(5) + 2(4)
11
4
23
6
c
Yeong-Chyuan
Chung, September 25, 2015
13. (12pts) Solve the following system of linear equations:
2a + 6b − 5c = 5
a + 3b + c + 7d = −1
3a + 9b − c + 13d + e = 1
Solution:




2 6 −5 0 0 5
1 3 0 5 0 0
f
 1 3 1 7 0 −1  rre
→  0 0 1 2 0 −1 
3 9 −1 13 1 1
0 0 0 0 1 0
Let b = s where s is any real number. Let d = t where t is any real number.
Then a = −3s − 5t, c = −1 − 2t, and e = 0.
Or (a, b, c, d, e) = (−3s − 5t, s, −1 − 2t,t, 0).
Problem Part 1
9
10
11
Score
7
12
13
Total