Honors Discrete DMTA MATRIX QUIZ REVIEW GUIDE - SOLUTIONS

1. A group of students is planning a retreat. They have contacted three lodges in the vicinity to inquire about rates. They found that Crystal Lodge charges $13.00 per person per day for lodging, $20.00 per day for food, and $5.00 per person for use of the recreational facilities. Springs Lodge charges $12.50 for lodging,$7.50 for use of the recreational facilities, and $19.50 for meals. Bear Lodge charges$18.00 a day for meals,$20.00 per night for lodging, and there is no extra charge for the recreational facilities.

Display this information in a matrix C where the rows are the lodges. Label the rows and columns.

2. Your math club is planning a Saturday practice session for an upcoming math contest. For lunch the students ordered 35 Mexican lunches, 6 bags of corn chips, 6 containers of salsa, and 12 six-packs of cold drinks. Suppose that the club pays $4.50 per lunch, $1.97 per bag of corn chips, $2.10 for each container of salsa, and $2.89 for each six-pack of cold drinks.

2a. Set up a matrix multiplication to find the total cost. Identify the rows and columns of your matrices.

2b. Find the total cost.

3. Mr. Jones has been shopping for a vacuum-powered cleaning system. He found one at Z-Mart and another model at Base Hardware. The Z-Mart system cost $39.50, disposal cartridges were 6 for $24.50, and storage cases were $8.50 each. At Base Hardware the system cost $49.90, cartridges were 6 for $29.95, and cases were $12.50 each.

3a. Write and label a matrix showing the prices for the three items at the two stores.

3b. Mr. Jones decided to wait and see if the prices for the systems would be reduced during the upcoming sales. When he went back during the sales, the Z-Mart prices were reduced by 10% and the Base Hardware prices were reduced by 20%. Construct a matrix showing the sale prices for each of the three items at the two stores.

3c. Use matrix subtraction to compute how much Mr. Jones could save for each item at the two stores.

4. An artist creates plates and bowls from small pieces of colored woods. Each plate requires 100 pieces of ebony, 800 pieces of walnut, 600 pieces of rosewood, and 400 pieces of maple. It takes 200 ebony pieces, 1200 walnut pieces, 1000 rosewood pieces, and 800 pieces of maple to make a large bowl. A small bowl takes 50 pieces of ebony, 500 walnut pieces, 450 rosewood pieces, and 400 pieces of maple.

4a. She currently has orders for five plates, three large bowls, and seven small bowls. Set up and solve a matrix multiplication to compute the number of pieces of each type of wood needed for the order.

4b. Next week she has one order for 6 plates, 2 large bowls, and 4 small bowls to ship out of town, and a second order for 12 plates and 9 small bowls for a local market. Set up and solve a matrix multiplication to compute the number of pieces of each type of wood needed to complete each order separately.

Soup / Sandwich
Restaurant #1 / $2.75 / $7.20
Restaurant #2 / $3.15 / $6.75
Restaurant #3 / $2.90 / $7.00
Restaurant #4 / $2.50 / $7.50

5. Four local restaurants offer all soups and sandwiches for a lunch special given in the table. On Monday, you order 5 soups and 4 sandwiches. On Wednesday, you order 2 soups and 7 sandwiches. On Friday, you order 6 soups and 10 sandwiches.

5a. Set up and solve a matrix multiplication to find the cost from each restaurant per day of the week.

Option 1:

Option 2:

5b. Which restaurant has the best deal per day of the week?

MONDAY = R#3 + R#4WEDNESDAY and FRIDAY = R#2

5c. If you can only order from one of four restaurants for the entire week, which restaurant should you choose?R#2 overall best

6. Three music classes at Central High are selling candy as a fundraiser. The number of each kind of candy sold by each of the three classes is shown in the following table.

Jazz Band / Symphonic Band / Orchestra
Almond Bars / 300 / 220 / 250
Chocolate Chews / 240 / 330 / 400
Mint Patties / 150 / 200 / 180
Sour balls / 175 / 150 / 160

The profit for each type of candy is sour balls, 30 cents; chocolate chews, 50 cents; almond bars, 25 cents; and mint patties, 35 cents. Use matrix multiplication to compute the profit made by each class on its candy sales.

7. The students at Central High are planning to hire a band for the prom. Their choices are A, B, and C.

They survey the Sophomore, Junior, and Senior classes and find the following percentages of students prefer the bands,

The student population by class and sex is:

Use matrix multiplication to find:

7a. number of males and females who prefer each band.

7b. total number of students who prefer each band.

8. The characteristics of a female reptile are shown in the following table. Suppose the initial female population for the heard is given by

Age Groups (yrs) / Birth Rate / Survival Rate
0 – 5 / 0 / 0.4
5 – 10 / 0.6 / 0.7
10 – 15 / 1.2 / 0.8
15 – 20 / 1.3 / 0.9
20 – 25 / 0.4 / 0.6
25 – 30 / 0 / 0

8a. What is the expected lifespan of this reptile?

30 YEARS

8b. Construct the Leslie matrix for this population.

8c. Find the new population distribution after 15 years? (Hint: How many cycles or transitions?)

15/5 = 3 transitions

8d. What is the total population of P6? TOTAL = 129

9. The characteristics of an insect are shown in the following table. Suppose the initial female population for the heard is given by

Age Groups (months) / Birth Rate / Survival Rate
0 – 4 / 0 / 0.8
4 – 8 / 0.7 / 0.5
8 – 12 / 0.9 / 0.6
12 – 16 / 1.2 / 0.7
16 – 20 / 0.3 / 0

9a. What is the expected lifespan of this insect?

20 MONTHS

9b. How long is a cycle or transition for this insect?

4 months long

9c. Construct the Leslie matrix for this population.

9d. What is the total population of P9?

TOTAL = 264

9e. Find the approximate number of female insects between 8 – 12 months in 5 transitions from now?