To find out how many coats, shirts, and slacks should be produced to use all available labor hours, we need to set up and solve a system of linear equations based on the given data. Here’s the step-by-step process to solve this problem:
1. Understand the Problem and Convert Units:
First, convert all available hours into minutes since the times for each process are given in minutes.
[tex]\[
\text{Available cutting time} = 115 \text{ hrs} \times 60 \text{ min/hr} = 6900 \text{ min}
\][/tex]
[tex]\[
\text{Available sewing time} = 280 \text{ hrs} \times 60 \text{ min/hr} = 16800 \text{ min}
\][/tex]
[tex]\[
\text{Available packaging time} = 65 \text{ hrs} \times 60 \text{ min/hr} = 3900 \text{ min}
\][/tex]
2. Set Up the System of Equations:
Let [tex]\( x \)[/tex] be the number of coats, [tex]\( y \)[/tex] be the number of shirts, and [tex]\( z \)[/tex] be the number of slacks.
We create the following equations based on the time requirements for each process per item:
Cutting Time Equation:
[tex]\[
20x + 15y + 10z = 6900
\][/tex]
Sewing Time Equation:
[tex]\[
60x + 30y + 24z = 16800
\][/tex]
Packaging Time Equation:
[tex]\[
5x + 12y + 6z = 3900
\][/tex]
3. Solve the System of Equations:
Solving this system of equations, we get:
[tex]\[
x = 120 \quad \text{(number of coats)}
\][/tex]
[tex]\[
y = 200 \quad \text{(number of shirts)}
\][/tex]
[tex]\[
z = 150 \quad \text{(number of slacks)}
\][/tex]
4. Interpret the Results:
Therefore, to utilize all available labor hours exactly, the clothing manufacturer should produce:
[tex]\[
120 \text{ coats}
\][/tex]
[tex]\[
200 \text{ shirts}
\][/tex]
[tex]\[
150 \text{ slacks}
\][/tex]
This way, the given labor hours for cutting, sewing, and packaging will be fully used without any surplus or deficit.
