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:
- Assembly constraint: 3x + 2y ≤ 24 (total assembly hours available)
- Painting constraint: 0.5x + y ≤ 8 (total painting hours available)
- Non-negativity constraint: x ≥ 0 and y ≥ 0 (number of units cannot be negative)
b. The objective function is to maximize profit, which can be represented as:
Profit = 15x + 12y
c. To graph the constraints, we can create 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 graph will have two lines:
- Assembly constraint: 3x + 2y = 24 (represented by a straight line)
- Painting constraint: 0.5x + y = 8 (represented by a straight line)
d. To find the vertices of the feasible region (the points where the lines intersect), we can solve the equations simultaneously:
3x + 2y = 24
0.5x + y = 8
Solving the equations, we get:
x = 4, y = 12 (point A)
x = 8, y = 0 (point B)
x = 0, y = 8 (point C)
e. To maximize profit, we need to determine the number of Deluxe models (x) and Economy models (y) at the vertices of the feasible region. From the vertices, we can see that:
- At point A (4 Deluxe models and 12 Economy models), profit = 15(4) + 12(12) = $240
- At point B (8 Deluxe models and 0 Economy models), profit = 15(8) + 12(0) = $120
- At point C (0 Deluxe models and 8 Economy models), profit = 15(0) + 12(8) = $96
Therefore, the Wellbuilt Company should produce 4 Deluxe models and 12 Economy models to maximize their profit.
f. The maximum profit would be $240.