Math 141 Exam 1 Review with Key

 

1. a)  Find the slope-y-intercept form of the line that passes through the point (-2,1) and has x-intercept 3.

b) For the line above, what is the change in y whenever x decreases by 4?

c) What is the change in x whenever y increases by 3?

 

2. Find the slope-int. form of the line through the point (0,4) given that each increase of 2 in x produces a decrease of 5 in y.

 

3. Find the general form of the line parallel to  and passing through (1, 4).

 

4. In a specific week, receipts at a theatre totaled $2882. Adult tickets were sold for $7 and child tickets were $5. The theatre sold three times as many child tickets as adult tickets. How many adult and how many child tickets were sold that week?

 

5. Joe wants to invest in a stock fund earning 12%, a mutual fund earning 7% and a bond fund earning 3%. Because the stock is riskier, he will put only 20% of the money into the stock fund. His total earning should come to $3900. Give the complete solution set to how much he should invest in each fund and also give two practical particular solutions.

 

6. Towit rental car company charges $25 per day plus gas. Their cars get 30 miles per gallon of gas. Jumpstart rental charges $0.50 per mile plus gas and their cars get 25 miles per gallon of gas. Gas is currently $2.70 per gallon. How many miles would you have to drive in a day for these charges to be equal? Which company is cheaper if you drive 70 miles?

 

7. A manufacturer has fixed costs of $6000 per month and each unit costs him an additional $6 to produce. If he sells each unit for $7.50, what are the cost, revenue and profit functions? What is the break even point?

 

8. A heifer was purchased for $1050 in the year 2000. It depreciated linearly to a value of 0 in 2007. What was the value in the year 2004?

 

9. Demand for a certain commodity is 500 units when the price is $325 and is 1500 units when the price is $275. Suppliers will provide 2000 units at a price of $400 per unit. For each $10 increase in price, they will provide 100 more units.

a) Find the demand equation. What is the lowest price for which none are demanded?

 

b) Find the supply equation. What is the highest price at which suppliers will provide none?

 

c) Find the equilibrium point.

 

10. An accounting firm gives an aptitude test to all its employees and compares the scores to the employees’ performance ratings. The table shows the aptitude score, x, and the performance rating, y, for five randomly selected employees.

 

            X=aptitude score          80   84    86    87    91

            Y=performance            7     8      9    8.7   9.5 

Find the linear regression equation and use it to predict the performance rating when

a) x=95   b) x=70  (Assume the performance rating can go above 10)

 

11.  A pollution control office is monitoring two waste treatment plants. They measure the amount of nitrates, lead and oxygen in the plant discharge in parts per million. The measurements at the beginning of the monitoring period are given by matrix E. The measurements at the end of the period are given by matrix F.

E=     F= What matrix represents the change in the amounts of pollutants in the discharge?

 

  1. The Blacks are each investing in two stocks. Matrix A shows the amounts each of them had in stocks 1 and 2 at the beginning of the year. Matrix B shows the amounts each had at the end of the year. If they owe 25% in taxes on the gains, what matrix shows their after tax gains in each stock for each person?

A=    B=

 

13. A=   B=   Find the entry in row 2 column 1 of AB.

14. For A and  B as in problem 13,  find .

 

15. Find the inverse to the coefficient matrix and use it to solve the system of equations   .

 

16. A police department employs two grades of personnel: rookies and sergeants. A rookie spends 20 hours training and 20 hours on patrol each week. A sergeant spends 5 hours training and 30 hours on patrol each week. The training center can handle 240 person hours each week while the department needs 440 person hours each week for patrol duty. How many persons at each grade should they have?

 

17. A chemistry department wants to make 2 liters of an 18% acid solution by mixing a 21% solution with a 14% solution. How many liters of each type of acid should they use?

 

 

 

18. Each augmented matrix represents a system of equations in x, y, and z. Find each solution or state none exists. If the solution is infinite, give the parametric solution and two particular solutions.

 

a)      b)     c)      d)

 

 

 

 

 

 

 

  1. Use rref in your calculator to solve each system of equations. If infinite, give the parametric solution and two particular solutions.

a)      b)      c)      

d)

 

  1. Each of the matrices is not fully row reduced. Perform the next pivot of the row reduction. Show all work and row operations.

 

a)     b)     c)

 

21. Perform the indicated operations for A, B, C.

 

 A=    B=      C=

a) A + B    b)     c) BC

22. A kitchen appliance manufacturer makes can openers and dough cutters using two machines, a press and a riveter. Each can opener requires 0.2 minutes in the press and 0.4 minutes in the riveter. A dough cutter requires 0.5 minutes in the press and 0.3 minutes in the riveter. The press can be operated only 3 hours per day and the riveter 2.5 hours per day. If these two machines are fully used, how many can openers and dough cutters can be produced each day?

 

23.  Write a product of 3 matrices to answer the following: A company produces two nut mixtures, regular and premium. The regular mix contains 60% peanuts, 25% cashews and 15% pecans. The premium mix contains 30% peanuts, 40% cashews and 30% pecans. Peanuts cost them $.80 per pound, cashews cost $1.15 per pound and pecans cost $1.50 per pound. How much will it cost them to fill an order for 4 pounds of regular mix and 12 pounds of premium mix?

 

24. The matrix shows the grades of 4 students, A, B, C and D on each of 3 quizzes.

 

     Q1   Q2   Q3

A

 5

7

8

B

9

10

9

C

8

8

7

D

6

10

9

Treat this as a matrix. what matrix would you multiply it by, and on which side to find:

a) the class average for each quiz.

b) the total quiz points for each student.

c) the quiz average for each student.

 

 

 

25. Which matrix operations can be performed for A=   B=   C=  If not possible, why not and if possible, what is the size of the resulting matrix?

 

a)  A+B    b) BC   c) AC    d)  CA       e) AB+C  

 

 

 

 

 

26. Which matrix has an inverse? Find the inverse when it exists.

 

A=      B=     C=

 

 

 

 

27.  Show that A=  and B=  are inverses and use this fact to solve

A= .   Your solution will contain the unknown, a.

 

28.  Solve the matrix equation AX+B=X  for X if  A=  and  B=.

29.  Solve each equation for X, given .

     a) AX = B + 3X

 

b) CX=B - 2X

 

 

 

 

 

 

 

 

30.  An economy is based on three sectors, agriculture, manufacturing, and energy.

Production of a dollar’s worth of agriculture requires inputs of $0.20 from each of the three sectors. Production of a dollar’s worth of manufacturing requires $0.40 from agriculture and $0.10 from each of the other two sectors. Production of a dollar’s worth of energy requires $0.30 from agriculture and $0.10 from each of the other two sectors. Find the output from each sector that is required to meet a demand of $10 billion for agriculture, $15 billion for manufacturing and $20 billion for energy.

 

Key:

 

1. a)

 

2.

 

3. 5x - 4y=-11

 

4. 131 adult and 393 child tickets

 

5. (9750+0.1b, 39000 – 0.6b, b) 0 < b < 65000  Two possible particular solutions are

b=0:  $9750 in stocks, 39000 in the mutual fund, and none in the bond fund

b=10000:   $10750 in stocks, $33000 in the mutual fund and $10000 in the bonds

 

6. They are equal at approximately 48.2625 miles. At 70 miles, Towit is cheaper.

 

7. (4000, 30000)

 

8. $450

 

9. a) p= -0.05x + 350   $350

 

b) p= 0.1x + 200    $200

 

c) (1000, 300)

 

10. a) 10.6141   b) 4.8319

 

11. E – F =

 

12. 0.75(B – A)=

 

  1. 2a + 20 + 7b

 

14.

 

15.      x = 5, y = 2, z = -5

 

16. 10 rookies, 8 sergeants

 

17. 1.143 liters of 21% and 0.857 liters of 14%

 

18. a) (3, -2, 5)    

 

b) ( -4 + z, -3-2z, z)  z can be any real number. Two particular solutions could be

z=0: ( -4, -3, 0)   z=1: ( -3, -5, 1)

 

c) no solution

 

d) ( -1, 3)

 

19. a) ( 4 + t,  -3-t, t ) t is any real number. Possible particular solutions are

t=0: ( 4, -3, 0) or  t=1: (5, -4, 1)

 

b) no solution

 

c)   t)  t any real number    t=0:  ( 3, 2, 0)    t=13: ( 8, 16, 13)

 

d)      ( 1, -2, 3)

 

20. a)  results in

b)    results in

c)   results in

 

21. a)      b)     

c)

 

22. 150 can openers and 300 dough cutters

23.     or       which both equal  $17.77

 

24.

 

 

 

 

 

 

 

 

 

 

25  a) not possible to add matrices of different sizes

 

b) 3x2     c) 2x2    d) 3x3   e) 2x3

 

26. A is not square so it cannot have an inverse.  B is square but is singular.  C has an inverse and C=

 

27. A and B are inverses since AB is equal to . ( BA is also).

 

28. Doing the algebra we have

29. a)     b)

 

30.  40.1 billion form agriculture, 29.4 billion from manufacturing and 34.4 billion from energy