A clothing manufacturer has 100m of silk and 180m of wool. To make a suit requires 2m of silk and 3m of wool, and to make a dress requires 1m of silk and 2m of wool. If the profit on a suit is $108 and the profit on a dress is $60, how many suits and dresses should the manufacturer make to maximize the profit?
To maximize the profit, we need to determine the number of suits and dresses the manufacturer should make using the given constraints.
Let's represent the number of suits as 'S' and the number of dresses as 'D'.
According to the information provided, each suit requires 2m of silk and 3m of wool, and each dress requires 1m of silk and 2m of wool.
To find the feasible range for 'S' and 'D', we use the following inequalities:
2S + D ≤ 100 (constraint for silk)
3S + 2D ≤ 180 (constraint for wool)
Since the manufacturer cannot produce a negative number of suits or dresses, we also have:
S ≥ 0 and D ≥ 0
The profit from producing 'S' suits and 'D' dresses is given by:
Profit = 108S + 60D
To maximize the profit, we can solve this problem using linear programming techniques. One common approach is the Simplex method, but we can also use graphing or substitution methods for this problem.
Let's solve it using the graphing method:
Step 1: Convert the inequalities into equations:
2S + D = 100
3S + 2D = 180
Step 2: Plot the constraints on a graph:
Plot the straight lines for each constraint using the intercept method or rearrange the equations to find the x and y-intercepts.
The graph should have labeled axes with S (number of suits) on the x-axis and D (number of dresses) on the y-axis.
The feasible region will lie within the intersection of the two lines, along with the non-negative quadrants.
Step 3: Determine the vertices of the feasible region:
Locate the points where the lines intersect. These are the vertices of the feasible region.
Step 4: Calculate the profit at each vertex:
Evaluate the profit function at each vertex by substituting the values of S and D into the profit equation.
Profit = 108S + 60D
Step 5: Find the vertex with the maximum profit:
Compare the profits calculated at each vertex and determine which vertex yields the maximum profit. The values of S and D at this vertex represent the optimal number of suits and dresses the manufacturer should produce to maximize the profit.
By following these steps, you can determine how many suits and dresses the manufacturer should make to maximize the profit.