5) A car manufacturer has manufacturing plants in 3 major cities. Below are the daily amounts of cars produced per plant for 3 different model types. (matrix A)

Model X
Model Y
Model Z
Plant A
12
10
0
Plant B
4
4
20
Plant C
8
9
12
Due to wage increases, the April profits are lower than the March profits. The profits are listed below. (matrix B)
March
April
Model X
$1000
$500
Model Y
$2000
$1200
Model Z
$1500
$1000
a. Calculate AB
b. Under the assumption that all the cars are sold, what was the daily profit in March from plant B?
c. What was the total daily profit (from all three plants) in April?

a. To calculate matrix AB, we need to multiply each element of matrix A by its corresponding element in matrix B and sum them up.

Matrix A:
Model X Model Y Model Z
Plant A 12 10 0
Plant B 4 4 20
Plant C 8 9 12

Matrix B:
March April
Model X $1000 $500
Model Y $2000 $1200
Model Z $1500 $1000

Calculating AB:
AB = (12 * $1000) + (10 * $2000) + (0 * $1500) (For Plant A, Model X)
+ (4 * $1000) + (4 * $2000) + (20 * $1500) (For Plant B, Model Y)
+ (8 * $1000) + (9 * $2000) + (12 * $1500) (For Plant C, Model Z)

AB = $12,000 + $20,000 + $0 + $4,000 + $8,000 + $30,000 + $8,000 + $18,000 + $18,000 = $118,000

Therefore, AB = $118,000

b. To calculate the daily profit in March from plant B, we need to find the sum of the profit for each model produced by plant B.

Matrix B:
March April
Model X $1000 $500
Model Y $2000 $1200
Model Z $1500 $1000

Daily profit in March from plant B = ($1000 * 4) + ($2000 * 4) + ($1500 * 20)
= $4,000 + $8,000 + $30,000
= $42,000

Therefore, the daily profit in March from plant B was $42,000.

c. To calculate the total daily profit from all three plants in April, we need to find the sum of the profit for each model produced by all three plants.

Matrix B:
March April
Model X $1000 $500
Model Y $2000 $1200
Model Z $1500 $1000

Total daily profit in April = ($500 * 12) + ($1200 * 9) + ($1000 * 12)
= $6,000 + $10,800 + $12,000
= $28,800

Therefore, the total daily profit from all three plants in April was $28,800.

To calculate AB, we need to perform matrix multiplication. Here are the steps:

Step 1: Create matrices A and B based on the given information:
Matrix A:
Model X Model Y Model Z
Plant A 12 10 0
Plant B 4 4 20
Plant C 8 9 12

Matrix B:
March April
Model X $1000 $500
Model Y $2000 $1200
Model Z $1500 $1000

Step 2: To calculate AB, multiply each element in each row of matrix A with the corresponding element in each column of matrix B, then take the sum of these products. The resulting matrix will be the same size as matrix A, with the rows representing the plants and the columns representing the months.

AB = [ (12 * $1000) + (10 * $2000) + (0 * $1500) (12 * $500) + (10 * $1200) + (0 * $1000)
(4 * $1000) + (4 * $2000) + (20 * $1500) (4 * $500) + (4 * $1200) + (20 * $1000)
(8 * $1000) + (9 * $2000) + (12 * $1500) (8 * $500) + (9 * $1200) + (12 * $1000) ]

Simplifying the expression, we get:

AB = [ 47000 18500
31000 19500
51000 22000 ]

So, the resulting matrix AB is:
March April
Plant A $47,000 $18,500
Plant B $31,000 $19,500
Plant C $51,000 $22,000

To calculate the daily profit in March from plant B, we look at the corresponding value in the matrix B. From the given information, the daily profit of Model Z in March from plant B is $1500. Therefore, the daily profit in March from plant B is $1500.

To calculate the total daily profit in April (from all three plants), we sum up the values in the right column of matrix AB. From the given information, the daily profits in April from plant A, plant B, and plant C are $18,500, $19,500, and $22,000 respectively.

Total daily profit in April = $18,500 + $19,500 + $22,000 = $60,000.

Therefore, the total daily profit in April (from all three plants) is $60,000.