The Bookholder Company manufactures two types of bookcases out of oal and walnut. Model 01 requires 5 board feet of oak and 2 board feet of walnut. Model 02 requires 4 board feet of oak and 3 board feet of walnut. A profit of $75 is made on each Model 01 bookcase and a profit of $125 is made on each Model 02 bookcase. The company has a supply of 1000 board feet of oak and 600 board feet of walnut. The company has orders for 40 Model 01 bookcases and 50 Model 02 bookcases. These order indicate the minimum number the company must manufacture of each model.
a. Write the set of constraints.
b. Write the objective function.
c. Graph the set of constraints.
d. Determine the number of bookcases of each type the company should manufacture in order to maximize profits.
e. Determine the maximum profit.
Please HELPPPPPPPPPP!!!
I will be happy to critique your thinking. Constrants: min number of each, board feet of lumber.
Objective function: I would max profit.
5x + 4y < or = 1000
2x + 3y < or = 600
4x + 5y > or = 9x > or = 40
y > or = 50
I have solve them and graphed them but I am at a loss for the last two questions in the problem. It seems that these only figure out the wood but I need to figure out the profit. That is where it lost me. Can you help me and tell me if I did this wrong? Or what to do?
You have correctly identified the set of constraints for the problem. The constraints are:
5x + 4y ≤ 1000 (constraint for oak)
2x + 3y ≤ 600 (constraint for walnut)
9x ≥ 40 (constraint for Model 01 bookcases)
y ≥ 50 (constraint for Model 02 bookcases)
Now, let's write the objective function. The objective is to maximize profit. The profit for each Model 01 bookcase is $75, and for each Model 02 bookcase is $125.
Let's assign the variables x and y to represent the number of Model 01 and Model 02 bookcases produced, respectively.
The objective function is:
Profit = 75x + 125y
Now, to determine the number of bookcases of each type the company should manufacture in order to maximize profits, we need to solve the system of linear inequalities and find the feasible region.
To do this, graph the set of constraints on a graph and shade the feasible region. The feasible region is the region that satisfies all the constraints.
Then, you can find the coordinates of the vertices of the feasible region and substitute each coordinate into the objective function to find the maximum profit.
Here's an example on how to solve this type of problem using a graph:
1. Graph the constraints:
- Plot the equations 5x + 4y = 1000 and 2x + 3y = 600 as two lines.
- Plot the inequalities x ≥ 40 and y ≥ 50 as vertical and horizontal lines, respectively, shading the region above and to the right of each line.
2. Shade the feasible region, which is the area where all the lines overlap/shade. This region represents the combinations of Model 01 and Model 02 bookcases that satisfy the constraints.
3. Find the vertices of the feasible region by identifying the points where the lines intersect.
4. Substitute the coordinates of each vertex into the objective function (Profit = 75x + 125y) to determine the profit at each vertex.
5. Compare the profits at each vertex and choose the vertex that results in the maximum profit. That will give you the number of bookcases of each type the company should manufacture.
6. Finally, substitute the coordinates of the vertex that maximizes profit back into the objective function to determine the maximum profit.
I hope this explanation helps you understand the process of solving this problem. If you still have any questions or need further clarification, feel free to ask!