A small tailors’ company wants to use at least 130 yards of fabric to sew evening skirts and dresses. A dress requires 4 yards of fabric and the production of a skirt will need 3 yards. Research shows that they will be able to sell at most three times as many skirts as dresses . A dress will take 10 hours to produce and a skirt will take 1 hour. They can assign to this work no more than 286 hours. Each dress will sell for $540, and each skirt will sell for $180. How many skirts should they sew to maximize the profit?
To determine the number of skirts the tailors' company should sew to maximize profit, we need to apply linear programming.
Let's define the variables:
Let D be the number of dresses to be sewn.
Let S be the number of skirts to be sewn.
Next, we need to establish the objective function. The objective is to maximize profit. The profit is the difference between the revenue and the cost. In this case, the revenue comes from selling dresses and skirts.
The revenue from selling dresses is calculated as: R_dresses = D * $540.
The revenue from selling skirts is calculated as: R_skirts = S * $180.
Now, we need to define the constraints:
1. The company wants to use at least 130 yards of fabric, so the fabric constraint is: 4D + 3S ≥ 130.
2. The company can assign no more than 286 hours of work, so the time constraint is: 10D + S ≤ 286.
3. The company can sell at most three times as many skirts as dresses, so the constraint is: S ≤ 3D.
To solve this linear programming problem, we can use graphical methods or algebraic methods, such as the simplex method. Let's use the graphical method.
To graph the constraints, we will first plot the equality lines for each constraint and then shade the feasible region.
1. Fabric constraint: 4D + 3S ≥ 130
Rearranging, we get: S ≥ (130 - 4D) / 3
Plotting the line S = (130 - 4D) / 3 on the graph.
2. Time constraint: 10D + S ≤ 286
Rearranging, we get: S ≤ 286 - 10D
Plotting the line S = 286 - 10D on the graph.
3. Skirt-to-dress constraint: S ≤ 3D
Plotting the line S = 3D on the graph.
The feasible region is the shaded area that satisfies all three constraints.
Next, we need to determine the corner points of the feasible region. We evaluate the objective function (profit) at each corner point to find the maximum profit.
The corner points can be obtained by solving the equations for the intersecting lines. In this case, we need to find the intersecting points of all three lines.
Once we have the corner points, we substitute the values into the objective function: profit = R_dresses + R_skirts = (D * $540) + (S * $180).
We evaluate the profit at each corner point and select the combination of skirts and dresses that yield the highest profit.
By following these steps, you can determine the number of skirts the tailors' company should sew to maximize profit.