In Class Exam 1 Review Math 141

1. Find the equation of the line parallel to 8x – 6y = 52 through (3,-1). What is the slope of this line?

2. Suppliers will supply none of a certain product if the unit price is $350 or less. They will supply 2000 if the price is $600. Find the linear supply equation.

3. Given two equations 30p + 2x = 18500 and 7p – x = 650 where p is price per unit and x is quantity, which is the supply equation and which is the demand equation? Find the equilibrium quantity and price. Show your work.

4. Consumers will buy none of a product at $750 per unit. For each decrease of $50 in the price, they will buy 200 more units. Find the price as a linear function of quantity. Is this the demand or the supply equation?

5. A company finds their fixed costs for a certain product total $21000. Each unit costs and additional $5 to produce and sells for $12. Find the break even quantity and the break even revenue.

6. Set up but do not solve the following word problem.

A nut mix contains peanuts, cashews, and pecans. Peanuts cost $1.50 per pound, cashews cost $2.50 per pound, and pecans cost $4.00 per pound. The mix contains 6 pounds and costs $2.75 per pound. There are half as many pounds of pecans as peanuts and cashews combined. How many pounds of each does the mix contain?

7. Each augmented matrix is completely reduced and represents a system of equations. Write the complete solution set for each. If the solution set is infinite, give two particular solutions.

i)

ii)

iii)

8.

a) Find

b) Find

9. 3x + 4y = 1

5x + 6y = b

Write the matrix equation for this system and solve it using an inverse matrix. Your answer will contain the unknown, b.

10. Demand quantities for a product at different prices are shown in the table.

Quantity demanded 2567 3479 4238 5798

dollars per unit 430 400 370 340

a) Find the linear regression equation for the price as a function of quantity.

b) According to the equation, at what price will the demand be 7000 units?

11. 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.

12. A meal contains foods A, B and C.. The meal weighs 6 ounces and contains 36 mg of iron. The iron contents per ounce of foods A, B and C are 10 mg, 8 mg and

2 mg respectively. How many ounces of each of A, B and C were used in the meal? Define variables clearly, write the equations and solve with rref. Give the parametric solution and two particular, realistic solutions.

13. Same as #12 but require twice as much of B as of A. Give the unique solution.

14. 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 then A is ___________.

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

15. To add the entries in each row (to add up each row) of an mxn matrix, multiply on the ______________ side by ________________.

To add the entries of each column (to add up each column) of an mxn matrix, multiply on the ___________ side by __________________.

16. 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.

17. The table shows the numbers of grams of protein, fat, and carbohydrate per ounce for three different foods.

almonds bread cheese

protein 6 3 8

fat 15 1 6

carb. 6 14 1

Protein has 4 calories per gram, fat has 9.5 calories per gram and carb. has 4 calories per gram. Write a product of three matrices to give the total number of calories in 2 ounces of almonds, 3 ounces of bread, and 4 ounces of cheese.

18. 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 time to mix, the time to bake and the time to package 4 cakes, 6 batches of cookies and 8 batches of brownies?

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

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

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

20. A= Use an inverse matrix to solve:

a) AX=

b) AX = + 4X

21. This is from the textbook by Tan.

An economy consists of three sectors: food, clothing and shelter.

The production of 1 unit of food requires the consumption of 0.4 units of food, 0.2 unit of clothing and 0.2 unit of shelter

The production of 1 unit of clothing requires the consumption of 0.1 unit of food, 0.2 unit of clothing and 0.3 unit of shelter.

The production of 1 unit of shelter requires the consumption of 0.3 unit of food, 0.1 unit of clothing and 0.1 unit of shelter.

Write the input-output matrix. How much production will satisfy the demand for $100 million of food, $30 million of clothing and $250 million of shelter?