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 an 15 cups of flour. When you sell these at the Farmers 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 profit.
Let x=The number of apple pies you make. Let y=The number of apple cobblers you make Write an inequality to show the constraint on the amount of apples you have.
1a. Apple pie uses 4 cups and 3 cups of flour.
Apple cobbler uses 2 cups of apples and 3 cups of flour
You have 16 cups of apples and 15 cups of flour
Apple pie= $3.00, $2.00 apple cobbler.
X=apple pie, Y=apple cobbler
Write an inequality to show the constraint on the amount of apples you have.
4x+2y<=16
1b. Write an inequality to show the constraint on the amount of flour you have.
3x+3y<=15
1c. Write any non-negativity constraints on x and y.
Add 4x+2y<=16 and 3x+3y<=15
7x+5y<=31
7x<=31, 5y<=31
2a. Leaving your inequality from 1a in standard form, find the x and y intercepts to graph it on the coordinate plane provided.
I need help with 2a. and 1c.
Explain it ?
Sure! I'd be happy to explain.
1c. Non-negativity constraint:
In linear programming, we usually assume that the variables, x and y in this case, cannot take on negative values. This is because in a real-world scenario, you can't make a negative number of apple pies or apple cobblers. Therefore, we can add the constraint that both x and y must be greater than or equal to zero.
So the non-negativity constraint is:
x ≥ 0
y ≥ 0
Moving on to 2a...
To find the x and y intercepts of the inequality from 1a, we can set one variable to zero at a time and solve for the other variable.
If we set x = 0 in 4x + 2y ≤ 16, we get:
0 + 2y ≤ 16
2y ≤ 16
y ≤ 8
So the y-intercept is (0, 8).
Now, let's set y = 0:
4x + 2(0) ≤ 16
4x ≤ 16
x ≤ 4
So the x-intercept is (4, 0).
To graph this inequality on the coordinate plane, plot the x-intercept at (4, 0) and the y-intercept at (0, 8). Then draw a straight line connecting these two points. Since the inequality is "less than or equal to", the line should be solid. You would shade the region below or on the line to represent all the possible combinations of x and y that satisfy the inequality.
I hope that helps! Let me know if you have any other questions.