If some one can get me started, I should be able to graph this.
MONTGOMERY COUNTY COMMUNITY COLLEGE
MAT106
SUMMER, 2007 ASSIGNMENT #1
NAME______________________________
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 $18 per unit and the profit on
the Economy model is $16 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? __________________
graph on two axis. right axis is assembly hours, up axis is paint hours.
So Deluxe intercepts is 8,0 and 0,16
The economy intercepts are 12,0 and 0, 8
I assume you can do the rest.
Sure! Let's break down the problem step by step:
a. List the constraints:
The constraints are the limitations that the company needs to consider in terms of assembly and painting hours. Based on the information given, we have the following constraints:
- Assembly hours constraint: The maximum number of assembly hours available is 24 per day. This constraint can be expressed as: 3x + 2y ≤ 24, where x represents the number of Deluxe models and y represents the number of Economy models.
- Painting hours constraint: The maximum number of painting hours available is 8 per day. This constraint can be expressed as: 0.5x + y ≤ 8.
b. Determine the objective function:
The objective function is the equation that needs to be maximized. In this case, we want to maximize the profit. The profit from selling Deluxe models is $18 per unit, and the profit from selling Economy models is $16 per unit. So the objective function can be expressed as: Profit = 18x + 16y.
c. Graph the set of constraints:
To graph the set of constraints, we need to plot the points of intersection between the constraints. We have two constraints: 3x + 2y ≤ 24 and 0.5x + y ≤ 8. Plotting these points will give us the feasible region.
d. Find the vertices of the feasible region:
The vertices of the feasible region are the points where the constraints intersect. These points represent the maximum and minimum values of the objective function within the given constraints. To find the vertices, simply solve the system of equations formed by the constraints. In this case, you have two equations: 3x + 2y = 24 and 0.5x + y = 8. Solve these equations simultaneously to find the vertices.
e. How many Deluxe models and how many Economy models should the Wellbuilt Company produce to maximize their profit?
Once you have the vertices of the feasible region, you need to substitute the coordinates of each vertex into the objective function (Profit = 18x + 16y) to find the maximum profit. The combination of Deluxe and Economy models that gives the maximum profit will be the answer.
f. What is the maximum profit?
Using the vertices obtained from the previous step, substitute the coordinates of each vertex into the objective function (Profit = 18x + 16y) to calculate the profit. Identify the vertex that gives the highest profit to find the maximum profit.
For the graph, you mentioned using two axes: the right axis represents assembly hours, and the up axis represents paint hours. The intercepts you mentioned for Deluxe and Economy models are correct.
Following these steps should help you solve the problem. Good luck!