MTH212

Unit 3 – Individual Project A

1.To raise money, the local baseball teams decided to sell team logo hats (H) and T-shirts (T).The league director decided to hold a contest among the teams to see which team can raise the most money.The contest lasted for 3 weeks.Here are the results of the first 2 weeks.The numbers represent the number of hats and T-shirts sold.

A.How much of each item had the teams sold by the end of the second week.Use matrices to solve the problem.Final answer must be given in matrix form.Show all work to receive full credit.

Add both the matrices:

Items sold at the end of week 2 =

B. Which team had sold the most items at the end of the second week, and how many total items did they sell?

Answer:

Team bears sold =25 + 55 = 80

Team chargers sold = 17 + 52 = 69

Team Tigers sold = 22 + 89 = 111

Team Blue Jays sold = 25 + 55 = 80

Thus team Tigers sold most items. Tigers sold 111 items.

C. By the end of the third week, the totalswere as follows:

The profit matrix will be .

Profit made by each team =

D.Which baseball team won the contest, and what was their total sales?

Team Tigers sold most items and made a profit of $201.

2.Use augmented matrices to solve the following systems of equations.Show all work to receive full credit.Final answer must be given in matrix form.

A.

Answer:

The augmented matrix will be

Perform the operation R1->(1/3)R1

Perform the operation R2->R2 – R1

Perform the operation R2->(3/5)R2

Perform the operation R1->R1 -(4/3)R2

This gives us x = 5, y = -1

Answer : x = 5, y = -1

B.

Answer:

The augmented matrix will be

Perform the operation R2 – 4R1

Which gives 0 = -7

Thus this system does not have any solution.

3. A company’s employees are working to create a new energy bar. They would like the two key ingredients to be peanut butter and oats, and they want to make sure they have enough carbohydrates and protein in the bar to supply the athlete.They want a total of 22 carbohydrates and 14 grams of protein to make the bar sufficient.Using the following table, create a system of two equations and two unknowns to find how many tablespoons of each ingredient the bar will need.Solve the system of equations using matrices.Show all work to receive full credit.

Carbohydrates

/

Protein

Peanut Butter / 2 / 4
Oats / 8 / 1

A.Write an equation for the total amount of carbohydrates.

Let x be the number of table spoons of peanut butter and y be the number of table spoons of oats.

2x + 8y = 22

B. Write an equation for the total amount of protein.

4x + y = 14

C. Determine the augmented matrix that represents the previous two equations.

Augmented matrix will be

D. Solve for the previous matrix. Show all work to receive full credit.

Perform the operation R1->(1/2)R1

Perform the operation R2->R2 – 4R1

Perform the operation R2->(-1/15)R2

Perform the operation R1->R1 – 4R2

Which gives x = 3, y = 2

E. How many tablespoons of each will there need to be for the new energy bar?

Number of table spoons of peanut butter = x = 3

and number of table spoons of oats = y = 2

4.A total of 700 tickets were sold for a musical.Senior citizen tickets sold for $15, children tickets sold for $20, and adult tickets sold for $25; the total earnings from ticket sales was $15,750. Five times more children tickets were sold than senior citizen tickets.How many tickets of each type were sold?Set up a system of three equations and three unknowns, use an augmented matrix to solve, and show all work to receive full credit.

A.What are the three unknowns?

Three unknowns are number of senior citizen tickets, number of children tickets and number of adult tickets.

Let x be the number of senior citizen tickets, y be the number of children tickets and z be the number of adult tickets.

B. Write a separate equation representing each of the first three sentences of the word problem.

A total of 700 tickets were sold for a musical.

x + y + z = 700

Senior citizen tickets sold for $15, children tickets sold for $20, and adult tickets sold for $25; the total earnings from ticket sales was $15,750.

15x + 20y + 25z = 15750

Five times more children tickets were sold than senior citizen tickets.

y = 5x

5x – y = 0

C. Determine the augmented matrix that represents the three equations.

The augmented matrix will be

D. Solve for the matrix.Show all work to receive full credit.

The augmented matrix will be

Perform the operation R2->(R2 – 15R1) and R3->(R3 – 5R1)

Perform the operation R2->(1/5)R2

Perform the operation R3->R3 + 6R2

Perform the operation R3->(1/7)R3

Perform the operation R2->R2 – 2R3

Perform the operation R1->R1 – R3

Perform the operation R1 – R2

This gives us x = 50, y = 250 and z = 400

E.How many of each type of ticket were sold?

Number of senior citizen tickets = x = 50

Number of children tickets = y = 250

Number of adult tickets = 400