the bookholder company maunfactures two types of bookcases out of oak 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 orders indicate the minimum number the company must manufacture of each moder.
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
c) determine the maximum profit
I would graph oak on the y axis, walnut on the x axis. Your constraints are you can not use negative amounts, and the maximum amounts are as given. Plot the two lines for model 1. At 1000bf oak, the point 400,1000 is on the model01 line, choose another point on that line (say 200,500) and draw the line. Do the same for model02. Start with 600,800 as one point, then the next point 300,400, draw the line.
Your objective is to maximize profit. Test the points on the outer edge corners of the figure. you have drawn.
Let me know how this works out.
I would graph oak on the y axis, walnut on the x axis. Your constraints are you can not use negative amounts, and the maximum amounts are as given. Plot the two lines for model 1. At 1000bf oak, the point 400,1000 is on the model01 line, choose another point on that line (say 200,500) and draw the line. Do the same for model02. Start with 600,800 as one point, then the next point 300,400, draw the line.
Your objective is to maximize profit. Test the points on the outer edge corners of the figure. you have drawn.
Let me know how this works out.
I am still confused as silly as that may sound. Math has never been my best subject. Can you give more details.
Of course, I can give you more details. Let's break down the problem step by step.
a) Write the set of constraints:
The constraints in this problem are the limitations on the supply of oak and walnut, as well as the minimum number of bookcases of each model that the company must manufacture.
Supply of oak:
The company can use up to 1000 board feet of oak.
Supply of walnut:
The company can use up to 600 board feet of walnut.
Minimum number of bookcases:
The company must manufacture at least 40 model 01 bookcases and 50 model 02 bookcases.
So the constraints can be written as follows:
Oak constraint: 5x + 4y ≤ 1000
Walnut constraint: 2x + 3y ≤ 600
Minimum bookcase constraint: x ≥ 40, y ≥ 50
Where x represents the number of model 01 bookcases and y represents the number of model 02 bookcases.
b) Write the objective function:
The objective is to maximize profits. We can express this as the total profit made from selling the bookcases.
Profit from model 01: $75 per bookcase
Profit from model 02: $125 per bookcase
So the objective function can be written as follows:
Objective function: Maximize Profit = 75x + 125y
c) Graph the set of constraints:
To graph the set of constraints, you can plot the lines that represent the equations of the constraints on a graph. In this case, plot the lines for each constraint and shade the feasible region where all the constraints are satisfied.
d) Determine the number of bookcases of each type the company should manufacture in order to maximize profits:
To determine the number of bookcases of each type the company should manufacture in order to maximize profits, you need to find the corner points of the feasible region (the shaded area in the graph). Evaluate the objective function at each corner point and choose the point that gives the maximum value as the solution.
e) Determine the maximum profit:
Once you have the number of bookcases of each type from step d, substitute these values into the objective function (75x + 125y) to calculate the maximum profit.
I hope this clarifies the steps involved in solving the problem. Let me know if you still have any questions!