2. The Witherspoon Down Company specializes in making down jackets, ski vests, and comforters. The requirements for down and labor and the profits earned are given in the following chart:
Down (pounds) Time (labor-hours) Profit ($)
Jacket 3 2 6
Vest1 2 1 6
Comforter 4 1 2
Each week the company has available 600 pounds of down and 275 labor-hours. It wants to earn a weekly profit of $1150. How many of each item should the company make each week?
To determine how many of each item the company should make each week, we can set up a system of equations using the given information.
Let's denote the number of jackets as 'x', the number of vests as 'y', and the number of comforters as 'z'.
Based on the information given in the chart, we can create the following equations:
Equation 1: 3x + 2y + 4z = 600 (equating the total down requirement to the available down)
Equation 2: 2x + y + z = 275 (equating the total labor-hour requirement to the available labor-hours)
Equation 3: 6x + 6y + 2z = 1150 (equating the desired weekly profit)
Now we can solve this system of equations to find the values for x, y, and z, representing the number of jackets, vests, and comforters, respectively.
There are multiple methods to solve this system of equations. One common method is substitution, but in this case, we will use a matrix-based method called Gaussian elimination.
Step 1: Write the augmented matrix for the system of equations:
| 3 2 4 | | x | | 600 |
| 2 1 1 | * | y | = | 275 |
| 6 6 2 | | z | | 1150 |
Step 2: Transform the augmented matrix into reduced row-echelon form.
Using row operations, subtract 2/3 times row 1 from row 3:
| 3 2 4 | | x | | 600 |
| 2 1 1 | * | y | = | 275 |
| 0 3 -2 | | z | | 250 |
Using row operations, subtract 2/3 times row 1 from row 2:
| 3 2 4 | | x | | 600 |
| 0 -1 -1 | * | y | = |-125 |
| 0 3 -2 | | z | | 250 |
Using row operations, add 3 times row 2 to row 3:
| 3 2 4 | | x | | 600 |
| 0 -1 -1 | * | y | = |-125 |
| 0 0 -5 | | z | | 875 |
Step 3: Solve for the variables using back substitution.
From the last row, we get z = 875/(-5) = -175.
From the second row, we get -y - z = -125. Plugging in the value of z, we have -y + 175 = -125 which gives y = 50.
From the first row, we get 3x + 2y + 4z = 600. Plugging in the values of y and z, we have 3x + 2(50) + 4(-175) = 600. Simplifying, we get 3x - 350 - 700 = 600, which gives 3x = 1650, and finally, x = 550.
Therefore, the company should make 550 jackets, 50 vests, and -175 comforters each week.
Note: The negative value for comforters indicates that the company should not produce any comforters to achieve the desired profit of $1150.