Question

A clothing manufacturer has 1,000 yd. of cotton to make shirts and pajamas. A shirt requires 1 yd. of fabric, and a pair of pajamas requires 2 yd. of fabric. It takes 2 hr. to make a shirt and 3 hr. to make the pajamas, and there are 1,600 hr. available to make the clothing. i. What are the variables? ii. What are the constraints? iii. Write inequalities for the constraints. iv. Graph the inequalities and shade the solution set. v. What does the shaded region represent? vi. Suppose the manufacturer makes a profit of $10 on shirts and $18 on pajamas. How would it decide how many of each to make? vii. How many of each should the manufacturer make, assuming it will sell all the shirts and pajamas it makes?

Answer 1

Let the number of shirts is x and the number of pairs of pajamas is y

Then the variables are x and y which are the numbers of shirts and pajamas

Since each shirt needs, 1 yard and a pair of pajamas needs 2 yards

Since there are 1000 yards to make them

Then the first inequality is

`\begin{gathered} (1)x+(2)y\leq1000 \\ x+2y\leq1000 \end{gathered}`

Since the time to make a shirt is 2 hours and the time to make a pair of pajamas is 3 hours

Since there are 1600 hours available, then

The second inequality is

`\begin{gathered} (2)x+(3)y\leq1600 \\ 2x+3y\leq1600 \end{gathered}`

Then let us answer the questions

i. The variables are x and y

ii. The constraints are 1000, 1600

iii. The inequalities are

`\begin{gathered} x+2y\leq1000 \\ 2x+3y\leq1600 \end{gathered}`

iv. Let us draw the graph

The red area represents the 1st inequality

The blue area represents the 2nd inequality

The area of the two colors is the area of the solutions of the 2 inequalities

V.

The shaded region represents the solution of the 2 inequalities, the numbers of shirts and pajams

Vi.

The intersection point between the 2 lines is (200, 400)

Then we will take this point to represents the number of shirts and pajamas

vii.

Since the profit on shirts is $10 and on pajama is $18

Then we should make 200 shirts and 400 pajamas