In Class Review for Exam 1  Math 166

 

  1. A carpenter and his son are building chairs, tables and cabinets. Each chair requires $20 worth of materials, 10 carpentry hours and 1 finishing hour. Each table requires $30 for materials, 12 carpentry hours and 3 finishing hours. Each cabinet requires $50 for materials, 16 carpentry hours and 4 finishing hours. They have $300 to spend, 120 hours for carpentry by the father and 21 hours for finishing by the son. How many of each should they build to use all their resources? Define variables clearly, write the equations and solve with rref.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

  1. A dog food contains chicken, soy and rice. Six ounces of dog food contains 36 grams of protein. The protein contents per ounce of chicken, soy and rice are 10 grams, 8 grams and 2 grams respectively. If there is twice as much soy as chicken, how much of each of chicken, soy and rice was used in six ounces of dog food? Define variables clearly, write the equations and solve with rref.

 

 

 

 

 

 

 

 

 

 

 

  1. Same as #2 but do not require twice as much soy as chicken. Give a realistic parametric solution and two possibilities.

 

 

  1. Solve for r, s, and t.  

 

 

 

 

 

 

 

 

 

  1. State the rules about matrix sizes for different operations.

Addition can be performed if and only if _________________.

 

 

The multiplication AB is possible if and only if ___________________.

 

If A is mxn the A is ___________.

 

If A has an inverse matrix then A is ____________  and _______________.

 

6. A small town voted on a bond proposal. 70% of republicans voted to pass the bond, 60% of democrats voted to pass it and 55% of independents voted to pass it. The total numbers of people who voted in each party were 552 republicans, 456 democrats and 98 independents. Write a matrix product that gives the total number of votes in favor of the bond.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

7. a) A person is making cakes, cookies and brownies. Each cake takes 10 minutes to mix, 60 minutes to bake and 2 minutes to package. Each batch of cookies takes 7 minutes to mix, 30 minutes to bake and 3 minutes to package. Each batch of brownies takes 10 minutes to mix, 20 minutes to bake and 2 minutes to package. What matrix product shows the total time to mix, bake and package 4 cakes, 6 batches of cookies and 8 batches of brownies? Write a product of 2 matrices.

 

 

 

 

 

 

 

 

 

 

 

b) If this were a company and the costs per minute to mix, bake and package are $.05,

$1.00, and $.03, what matrix product shows the total cost of 4 cakes, 6 batches of cookies and 8 batches of brownies?

 

 

 

 

 

 

 

8. You should also be able to completely reduce a 2x2 system of equations using Gauss-Jordan either in the equations or the augmented matrix.

Perform the next pivot in the row reduction of the augmented matrix shown. Label each row operation.  

 

 

 

 

 

 

 

 

 

 

 

9. A=  Use an inverse matrix to solve:

a) AX=

 

 

 

 

 

 

 

 

b) AX-5X=   Hint: You need to factor out X.

 

 

 

 

 

 

 

 

 

 

10. State the conditions for a matrix to be in rref. Be able to tell what is wrong if it is not.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

11. An economy is based on three sectors, agriculture, manufacturing and energy. Production of one unit of agricultural products requires inputs of 0.3 unit from agriculture, 0.2 unit from manufacturing, and 0.3 unit from energy.

Production of one unit of manufactured products requires inputs of  0.3 unit from manufacturing, and 0.4 unit from energy.

Production of one unit of energy requires inputs of  0.2 from each sector.

a)      How much should be produced to meet a demand for 15 million units of agricultural goods, 10 million units of manufactured products and 8 million units of energy?

b)      How much of each sector is used up by production of the quantities from a)?

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

12. p is the proposition “I will study” , q is “I will do the review problems” and r is “I will pass the test”.

a)      Translate into words the statement .

 

 

 

 

 

b)      Write with symbols and make a truth table for “I will study and do the review problems or I will not pass the test”.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

13. True or False   Use Venn diagrams to support your answer.:

 

a) .

 

 

 

 

 

 

 

 

 

c)      If  then .

 

 

 

 

 

 

 

 

 

 

14. The universal set is the people in College Station.

T is all those in C.S who got a traffic ticket within a year.

A is all those in C.S. who had a traffic accident within a year.

S is all those in C.S. who have gone over the speed limit within a year.

 

Describe each set in words and shade it in a Venn diagram.

 

a)

 

 

 

 

 

 

 

 

 

b)

 

 

 

 

 

 

 

 

c)

 

 

 

 

 

 

 

 

15. A company is studying the effectiveness of its advertising. 45 people who use the product were asked if they read the newspaper ad or watched the TV ad.  Eight said they had done neither. A total of  20 read the newspaper ad and a total of 32 watched the TV ad. How many did both?

 

 

 

 

 

 

 

16. A, B, and C are three subsets of a universal set U. Make a Venn diagram and determine how many elements are in each region using the following information about how many elements belong to different sets:

 

10 belong to none of the three sets A, B, and C.

 

6 belong to A only.

 

7 belong to A but not C.

 

14 belong to either A or C, but do not belong to B.

 

23 belong to A or C, but do not belong to B.

 

5 belong to all three of A, B and C.

 

8 belong to B and C.

 

13 belong to B.