UMD Math 1160 Exam 2

Solve This

100

Given are matrices that have already been row reduced, write the solution to each system in the blank. If a solution does not exist, write no solution.

1  0  -1 | -5                1  0  0  | 3
0  1  3  |11                 0  1  0  | 4
0  0  0  | 0                  0  0  0  | -2

x=-5+z, y= 11-3z, z=z and no solution

200

-3x+5y=-2

-2x-y=3

x=-1, y=-1

300

-3x+5y=-1

-2x-y=6

x=-29/13, y= -20/13

400

x-y+5z=-6

3x+3y-z=10

x+2y+3z=5

x=-1/6, y= 10/3, z=-1/2

500

Solve the system using "Inverse of the Coefficient Matrix", that is, X= A-inverse * B.

-3x+5y=-2

-2x-y=3

x=y=-1

True or Not True

100

You can add or subtract any two matrices, just do it component-wise.

Not true, matrices must be the same dimensions, too.

200

If a feasible region is bounded on all sides then we will always have a maximum value and it will occur at a corner point.

This is True. (What about min values? What if the region is not bounded?)

300

In order to multiply two matrices A and B to get AB, the number of rows in A must be same as the number of rows in B.

False! (number of columns in A = number of rows in B, also what is the sized of the result?)

400

If for two matrices the product AB=1, then it must be that B is equal to A-inverse.

Possible, but not always true->False! For inverses, we need to be talking about square matrices, and, A*B=I, the identity matrix.

500

There will be a question on the test that will ask you to find the inverse of a 3 by 3 matrix.

This is true. Seriously.

Don't Solve This

100

800 people attend a basketball game, and total ticket sales are $3102. If adult tickets are 6 dollars and student tickets are 3 dollars, how many adults and students attended the game?

x+y=800, 6x+3y=3102

200

The Duluth Huskies play at Wade Stadium, which has 4,200 seats. Box seats cost $9, general admission tickets cost $7, and kennel club tickets cost $27. When all seats are sold, the revenue is $38,900. If the number of box seats is one-third the number of general admission seats, how many seats of each type are there?

x+y+z=4200

9x+7y+27z=38900

x-1/3*y=0

300

Certain lab animals must have at least 30 grams of protein and at least 20 grams of fat per feeding period. These nutrients come from food A, which costs $0.18 per unit and has 2 g. of protein and 4 g. of fat, and food B, with 6 grams of protein and 2 g. of fat costing $0.12 per unit. Food B is bought under a long term contract requiring that at least 2 units of B be used per serving. How much of each food must be bought to produce the minimum cost per serving.

Minimize: z=.18x+.12y
subject to: 2x+6y>=30
                4x+2y>=20
               x>=0, y>=2

Where x= amount of Food A and y= amount of Food B

400

An investor has $24,000 to invest in bonds of AAA and B qualities. The AAA bonds yield an average of 6%, and the B bonds yield 10%. The investor requires that three times as much money should be invested in AAA bonds as in B bonds. How much was invested if the total investment after one year was worth $27,200

x+y=24000

1.06x+1.10y=27200

x-3y=0

where "x=$ invested in AAA" and "y=$ invested in B"

500

Ichiro Suzuki holds the American League record for the most hits in a single baseball season. In 2004, Suzuki had a total of 262 hits for the Seattle Mariners. He hit three fewer triples than home runs, and he hit three times as many doubles as home runs. Suzuki also hit 45 times as many singles as triples. Find the number of singles, doubles, triples, and home runs hit by Suzuki during the season. Make sure to define your variables x, y, z, and w. (Source: Baseball Almanac)

Let x=# of singles, y=# of doubles, z= triples, w= homeruns.

x+y+z+w=262

-z+w=3

y-3w=0

x-45z=0

 

Straight Outta #2016Exams

100

Solve the following system of equations using any method. 5x+y=4, 3x+2y=1

(x, y)= (1, -1)

200

Solve the following system of equations: hint : A^(-1)=[(1.5, 0.5, -1.5), (-1, 0, 1), (0.5, -0.5, 0.5)].x+y+z=2, 2x+3y=5, x+2y+z=-1.

(x, y, z)=(7, -3,-2)

300

SET UP BUT DO NOT SOLVE THE FOLLOWING SYSTEM OF EQUATIONS PROBLEM: Annie wants to invest $100,000 in order to produce an annual income of $4,400. An investment service, Tibbers Financial Advising, recommends that she invest in treasury bonds that earn 4% annually, municipal bonds that earn 3.5% annually and corporate bonds that earn 5% annually. As a risk control factor, they recommend that the amount invested in corporate bonds should equal the total of the amounts invested in treasury bonds and municipal bonds. If Annie follows her advisor’s recommendations, how much should she invest in each of the three types of bonds?

x+y+z=100000

x+y-z=0

.04x+0.035y+0.05z=4400

400

Find the inverse of the following matrix:

          1  0  1
          2  -2  -1
          5  0  0

The inverse is:

              0  0  1/5
            -.5 -.5  .3
             1  0  -1/5

500

You are thinking of making your home more energy efficient by replacing some of the light bulbs with compact fluorescent bulbs, and insulating part or all of your exterior walls. Each compact fluorescent light bulb costs $4 and saves you an average of $2 per year in energy costs, and each square foot of wall insulation costs $1 and saves you an average of $0.20 per year in energy costs. Your home has 60 light fittings and 110 sq. ft. of uninsulated exterior wall. You can spend no more than $1200 and would like to save as much per year in energy costs as possible. How many compact fluorescent light bulbs and how many square feet of insulation should you purchase? How much will you save in energy costs per year?

Maximize Savings= 2x+0.2y
                            0<=x<=60
                            0<=y<=110
                            4x+y<=1200

Max: Purchase 60 light bulbs and 110 sq. ft. of insulation to save $142/yr.