An apple pie uses 4 cups of apples and 3 cups of flour. An apple cobbler uses 2 cups of apples and 3 cups of flour. You have 16 cups of apples and 15 cups of flour. When you sell these at the Farmer’s Market you make $3.00 profit per apple pie and $2.00 profit per apple cobbler. Use linear programming to determine how many apple pies and how many apple cobblers you should make to maximize your profit.
1a. Let x= the number of apple pies you make and y= the number of apple cobblers you make. Write an inequality to show the constraint on the amount of apples you have. (1 pt)
4x+2y<=16 (my answer)
1b. Write an inequality to show the constraint on the amount of flour you have. (1 pt)
3x+3y<=15 (my answer)
1c. Write any non-negativity constraints on x and y. (1 pt)
Note: For these next problems, there is a standard coordinate plane but it won't let me paste a picture.
2a. Leaving your inequality from 1a in standard form,
find the x and y intercepts to graph it on the coordinate
plane provided. Be sure to shade accordingly. (1 pt)
2b. Leaving your inequality from 1b in standard form,
find the x and y intercepts to graph it on the coordinate
plane provided. Be sure to shade accordingly. (1 pt)
3a. List the ordered pairs that represent the corners of your feasible region. (1 pt)
3b. Write an objective function that you can use to evaluate your profit P. (1 pt)
3c. Evaluate each ordered pair in your objective function. (1 pt)
3d. How many apple pies and how many apple cobblers should you make to maximize your profit? What is your maximum profit? (2 pts)
These next questions are bonus, they do not have to be answered. But if you'd like to show me how to do them as well, I'd appreciate it!
BONUS:
a. Rewrite your inequality from 1a in slope-intercept form. (+1)
b. Rewrite your inequality from 1b in slope-intercept form. (+1)
c. Graph the system in your graphing calculator. Find the corners of the feasible region. Include a screenshot of your graph below. (+2)
Any help is seriously appreciated! I am very desperate right now. I understand this is a lot to ask.
We do not do your homework for you. Although it might take more effort to do the work on your own, you will profit more from your effort. We will be happy to evaluate your work though.
Hey PsyDAG,
I understand this. And I honestly appreciate this policy. But to be straight-up with you, I have no idea how to even start. I am very glad that you would be willing to look over my work--but the thing is I have no work to show due to the fact I have no clue what I am doing. Thanks anyways.
1a. The inequality to show the constraint on the amount of apples you have is:
4x + 2y <= 16
Your answer is correct.
1b. The inequality to show the constraint on the amount of flour you have is:
3x + 3y <= 15
Your answer is correct.
1c. The non-negativity constraints on x and y are:
x >= 0
y >= 0
2a. To find the x-intercept, set y = 0 and solve for x:
4x + 2(0) = 16
4x = 16
x = 4
So the x-intercept is (4, 0).
To find the y-intercept, set x = 0 and solve for y:
4(0) + 2y = 16
2y = 16
y = 8
So the y-intercept is (0, 8).
Plotting these points on the coordinate plane and shading the appropriate region, we get:
|
_________|
y-intercept | shaded region
(0, 8) |
2b. To find the x-intercept, set y = 0 and solve for x:
3x + 3(0) = 15
3x = 15
x = 5
So the x-intercept is (5, 0).
To find the y-intercept, set x = 0 and solve for y:
3(0) + 3y = 15
3y = 15
y = 5
So the y-intercept is (0, 5).
Plotting these points on the coordinate plane and shading the appropriate region, we get:
|
________|
y-intercept | shaded region
(0, 5) |
3a. To find the corners of the feasible region, we need to find the intersection points between the lines represented by the inequalities.
The x-intercept of the first inequality is (4, 0), and the x-intercept of the second inequality is (5, 0). The y-intercept of the first inequality is (0, 8), and the y-intercept of the second inequality is (0, 5).
So the corners of the feasible region are (4, 0), (5, 0), (0, 8), and (0, 5).
3b. The objective function that represents the profit is:
P = 3x + 2y
3c. Evaluating each ordered pair in the objective function:
- For (4, 0):
P = 3(4) + 2(0) = 12 + 0 = 12
- For (5, 0):
P = 3(5) + 2(0) = 15 + 0 = 15
- For (0, 8):
P = 3(0) + 2(8) = 0 + 16 = 16
- For (0, 5):
P = 3(0) + 2(5) = 0 + 10 = 10
3d. To maximize the profit, we need to find the point with the highest value of P. From the evaluations above, we can see that the point (0, 8) gives the maximum profit of $16. Therefore, you should make 0 apple pies and 8 apple cobblers to maximize your profit.
The maximum profit is $16.
BONUS:
a. The inequality from 1a in slope-intercept form:
2y <= 16 - 4x
y <= 8 - 2x
b. The inequality from 1b in slope-intercept form:
3y <= 15 - 3x
y <= 5 - x
c. Unfortunately, as a text-based AI bot, I am unable to show images or screenshots. However, you can graph the system of inequalities on an online graphing calculator or graphing tool to find the corners of the feasible region. Simply plot the lines represented by the inequalities and locate the intersection points.