Math 125 Spring 2016 Lowman Written HW #1: Review for Exam I

classid: Required (missing or incorrect !5 points).

Name

TA Name and Discussion Time:

Write all answers in the provided exam booklet. Show all work for full credit.

Score = quiz score (8pts) + TA points (2pts) = Total (10pts)

Due on Monday after the exam at the beginning of the lecture. This page must be printed and stapled to

the other pages that contain your work. If pages are not stapled and this page is not attached then our HW

will not be graded or recorded. HW cannot be submitted by email. If you turn in your HW on Wednesday

or Friday in lecture of the same week it is due then it may be accepted for at most half credit. Homework

placed in my mailbox will not be accepted.

1. Graph the feasible set for the inequalities below, properly shaded in the Math 125 way.

8

><

>:

2x + 3y ? 15

y # 2(x ! 1)

x # 0, y # 0

2. Deterine by hand, where the following graphs intersect one another.

(

2x + 3y ? 15

y # 2(x ! 1)

3. Consider the following system of linear equations:

(

2x + y = 5

x + 4y = 6

a) write the corresponding augmented matrix;

b) reduce to diagonal form by hand using Gauss-Jordan elimination;

c) state the solution to the system of equations.

4. Consider the data points given in the table below:

x y xy x2

1 2

2 5

3 11

^x = ^y = ^xy = ^x2 =

a) plot theses points on the plane;

b) find the line of best-fit by hand using the formula;

c) graph the line obtained in b).

5. Use the Gauss-Jordan Elimination Method to solve each of the following systems of equations.

For each system you must:

• write the corresponding augmented coe!cient matrix;

• use the three elementary row operations to reduce to RREF.

• show all work as was done in lecture examples.

• state the solution (if one exists) to the system of equations.

(a) (

2x + y = 5

x + 2y = 4

(b) (

2x + 2y = 6

x + y = 2

(c) (

2x + y = 3

4x + 2y = 6

6. Given the matrix equation A · X = X ! D, use matrix algebra to solve for matrix X in terms of A

and D. Show all work.

7. Use the Gauss-Jordan method to find A!1, the inverse of A. Show all work.

A =

2

4

011

101

111

3

5

8. Given

8

><

>:

y + z = 3

x + z = 2

x + y + z = 4

(a) Write the system of equations in the matrix form A · X = B.

(b) Use the inverse matrix method to solve the system of equations.

(c) Solve the system of equations using the gauss-jordan method (Do not use the inverse matrix).

9.

8

Maximize the objective function z = 2x + 5y subject to the constraints

>>>>>><

>>>>>>:

x + 2y ? 20

3x + 2y # 24

x ? 6

x # 0

y # 0

(a) Graph the area of feasible solutions.

(b) Find all corner points.

(c) Find (x, y) that maximizes z and give the maximum value of z.

10. A simplified economy consists of the two sectors Transportation and Energy. For each $1 worth of

output, the transportation sector requires $.25 worth of input from the transportation sector and $.20

input from then from the energy sector. For each $1 worth of output, the energy sector requires $.30

from the transportation sector and $15 from the energy sector. You can use a calculator for (c) and

(d).

(a) Use matrix algebra to solve the matrix equation for X. You must show all steps. If you only give

the solution without showing all required algebra steps then you will not receive any credit.

X = A · X + D

(b) Give the input-output matrix A for this economy.

(c) Determine the matrix (I ! A)!1. (Round entries to two decimal places.)

(d) At what level of output should each sector produce to meet a demand for $5 billion worth of

transportation and $3 billion worth of energy?

11. A home appliance manufacturer has been selling a kitchen stove model several markets and wishes to

enter new markets. Atlanta has 6.1 million people and they sold 13,286 stoves, Tampa has 2.8 million

with 5,123 sold, Miami has 6.4 million with 17,522 sold, Charlotte(NC) has 2.5 million with 4,848 sold,

and Greenville(SC) has 1.4 million with 3,613 sold.

(a) Using Population figures in Millions and sales as they are given, use LinReg in constructing the

Least

(b) Squares Line of Best Fit for this data set. Round-o? values to the nearest thousandth.

(c) Use your line to predict the sales generated by entering a market like Orlando with 2.9 million

people.

(d) What sized market does your model suggest is necessary to sell 20,000 units?

12. Foodie Corp. makes food additives providing vitamins A and D. Each ounce of Type I has 200mg of

vitamin A, 100mg of vitamin D, and costs $0.24, while each ounce of Type II has 120mg of vitamin

A, 300mg of vitamin D, and costs $0.18 per ounce. A customer wants to create a vitamin supplement

granola bar with at least 800mg of vitamin A and at least 1200mg of vitamin D, but at the least cost.

Set up the Linear Programming Problem. DO NOT SOLVE.

13. Three-Sector Economy In an economic system, each of three industries depends on the others for raw

materials.

• To make $1 of processed wood requires:

– $.30 wood,

– $.20 steel

– $.10 coal.

• To make $1 of processed steel requires:

– $.00 wood,

– $.30 steel $.20 coal.

• To make $1 of processed coal requires:

– $.10 wood,

– $.20 steel

– $.05 coal.

To allow for $1 consumption in wood, $4 consumption in steel, and $2 consumption in coal, what levels

of production for wood, steel, and coal are required?