The Wellbuilt Company produces two types of wood chippers, Deluxe and Economy.
The Deluxe model requires 3 hours to assemble and ½ hour to paint, and the Economy
model requires 2 hours to assemble and 1 hour to paint. The maximum number of
assembly hours available is 24 per day and the maximum number of painting hours
available is 8 per day. If the profit on the Deluxe model is $15 per unit and the profit on
the Economy model is $12 per unit, how many units of each model will maximize profit?
Let x = number of Deluxe models
y = number of Economy models
a. List the constraints
b. Determine the objective function. __________________
c. Graph the set of constraints. Place number of Deluxe models on the horizontal axis
and number of Economy models on the vertical axis.
d. Find the vertices of the feasible region.
Vertices Profit
e. How many Deluxe models and how many Economy models should the Wellbuilt
Company produce to maximize their profit?
Deluxe____________
Economy__________
f. What is the maximum profit? __________________
a. The constraints for this problem are as follows:
1. Assembly constraint: 3x + 2y ≤ 24 (maximum number of assembly hours available)
2. Painting constraint: 0.5x + y ≤ 8 (maximum number of painting hours available)
3. Non-negativity constraint: x ≥ 0 and y ≥ 0 (number of units cannot be negative)
b. The objective function is to maximize profit. The profit is given by:
Profit = 15x + 12y
c. To graph the set of constraints, we can plot the feasible region on a graph with the number of Deluxe models (x) on the horizontal axis and the number of Economy models (y) on the vertical axis. The feasible region is the intersection of the shaded areas below the lines:
1. Assembly constraint: 3x + 2y ≤ 24
This line has intercepts:
- x = 0: 2y ≤ 24, y ≤ 12
- y = 0: 3x ≤ 24, x ≤ 8
Plot the line x = 8 (vertical), passing through (8, 0), and the line y = 12 (horizontal), passing through (0, 12). Shade the region below this line.
2. Painting constraint: 0.5x + y ≤ 8
This line has intercepts:
- x = 0: y ≤ 8
- y = 0: 0.5x ≤ 8, x ≤ 16
Plot the line x = 16 (vertical), passing through (16, 0), and the line y = 8 (horizontal), passing through (0, 8). Shade the region below this line.
d. To find the vertices of the feasible region, we need to identify the points where the lines intersect. These points are the vertices of the feasible region.
e. The feasible region may have multiple vertices. We need to evaluate the profit at each vertex to determine the optimal number of Deluxe and Economy models to maximize the profit. The coordinates of the vertices will help us find the solutions.
f. Once we have determined the optimal number of Deluxe and Economy models, we can calculate the maximum profit by plugging those values into the profit function: Profit = 15x + 12y.