Question

A clothing manufacturer has 1, 000yd. of cotton to make shirts and pajamas.

A shirt requires 1yd. of fabric, and a pair of pajamas requires 2yd. of fabric. It takes 2hr. to make
a shirt and 3hr. to make the pajamas, and there are 1, 600hr. available to make the clothing.
(i) What are the variables? Number of shirts made and number of pajamas made.
(ii) What are the constraints? How much time the manufacturer has and how much material is available.
(iii) Write inequalities for the constraints. Let x= number of shirts, and let y= number of pajamas. x≥0 and y≥0,x+2y≤1000,2x+3y≤1600.
(iv) Graph the inequalities and shade the solution set.

Answer 1

This question in mathematics involves formulating and graphing linear inequalities to find the feasible number of shirts and pajamas a manufacturer can produce given the fabric and time constraints. The graph's shaded region will depict the solution set that satisfies both constraints.

Step-by-step explanation:

The student's question deals with a problem of linear programming, where the objective is to determine the combination of shirts and pajamas that can be produced given certain constraints. The constraints are derived from the limited resources available in terms of fabric and production time. The problem is to be visualized using a graph where the feasible region represents the solution set that satisfies all constraints.

Variables

The variables in this scenario are the number of shirts (x) and the number of pajamas (y).

Constraints

The constraints include:

• The non-negativity constraint: x≥0 and y≥0,
• The fabric constraint: x+2y≤1000,
• The time constraint: 2x+3y≤1600.

Graphing the inequalities

To graph these inequalities:

1. Plot the equation x+2y=1000 on a graph as a straight line. This represents the fabric constraint.
2. Plot the equation 2x+3y=1600 on the same graph as another straight line. This represents the time constraint.
3. Identify the region where both constraints overlap. This is the feasible region.
4. Shade the feasible region which represents all the possible combinations of shirts and pajamas that the manufacturer can produce within the resource constraints.