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.
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. If anyone could help me I'd be so grateful. I been struggling with this for 2 days. I am not just looking for quick answers I want to learn. If you have any links that could help with constraints that would be amazing.
Let's take this step by step.
1. We can let x = number of apple pies you would make and y= the number of apple cobblers you make.
2. We can make an inequality showing the constraint of apples we need. So: 4x+2y<=16. Done!
3. Next ingredient we would use is flour, so let's make an inequality for the amount of flower we have: 3x+3y<=15. Just to help explain how we got it
4. We need 4 cups of apples for an apple pie and we need 2 cups of apples for an apple cobbler. Same applies with the flour, we ned 3 cups of flour and 3 cups of flour for both the apple pie and flour.
5. So, next we can graph the equations we wrote, so graph that and then write down what the feasible region is. (Should be three points).
6. Now, we can write the maximum profit equation, which would be 3x+2y. (Reason why is because $3 per apple pie and $2 per apple cobbler). For this you can solve it yourself and just plug into what we found when we graphed it.
Consider a duck swimming under water to catch a fish. His path can be written by the function . There is a piece of seaweed on the y-axis. X is time in seconds related to when he hits the seaweed. Y is his distance in feet under the water. Find and label each ordered pair. Match each vocabulary term with each moment (terms may be repeated).
2a. root (-10,0)
2b. vertex (-3,-49)
2c. y-intercept (0,-40)
2d. root (4,0)
Well, well, well! It looks like we have a math problem cooking here! Let's use linear programming to whip up some profits, shall we?
Let's start by defining our variables:
x = number of apple pies
y = number of apple cobblers
Now, let's set up some constraints based on the given information:
For the apples constraint, we have:
4x + 2y ≤ 16
And for the flour constraint, we have:
3x + 3y ≤ 15
Remember, these constraints ensure that you don't use more apples or flour than you have. We don't want any kitchen disasters, right?
Next up, we need to make sure we maximize our profits. Each apple pie brings in $3.00 profit, and each apple cobbler yields $2.00 profit. So, our objective function (profit) is:
P = 3x + 2y
Awesome! We've got everything laid out. Now, let the linear programming magic unfold.
To find the optimal solution, we need to graph the feasible region formed by the two constraints. From that region, we'll find the vertex that gives us the maximum profit.
To make things easier, let's first solve each constraint for y:
y ≤ (16 - 4x) / 2
y ≤ (15 - 3x) / 3
Now, let's plot these inequalities on a graph and find the feasible region. The vertex that gives us the highest profit is the optimal solution.
Unfortunately, as a text-based bot, I can't exactly draw it out for you, but I'm sure with a little effort, you can sketch it on some graph paper or use software that can do it for you.
Once you find the feasible region and the vertex providing the maximum profit, you'll have the values of x and y that will yield the greatest results.
And remember, even if you don't get it right on the first try, don't be too hard on yourself. After all, mistakes make the best ingredients for learning!
Good luck, my mathematical chef!
To solve this problem using linear programming, we need to set up the objective function and the constraints.
Objective Function:
We want to maximize the profit, so the objective function can be written as:
Profit = 3x + 2y
Constraints:
1. The amount of apples used in apple pies and cobblers cannot exceed the available amount:
4x + 2y ≤ 16 (apples constraint)
2. The amount of flour used in apple pies and cobblers cannot exceed the available amount:
3x + 3y ≤ 15 (flour constraint)
3. The number of apple pies and apple cobblers must be non-negative:
x ≥ 0 (non-negativity constraint for apple pies)
y ≥ 0 (non-negativity constraint for apple cobblers)
To graphically solve this problem, we can plot the feasible region and find the optimal solution.
First, let's simplify the constraint inequalities:
Apples constraint:
4x + 2y ≤ 16
Divide by 2:
2x + y ≤ 8 (revised apples constraint)
Flour constraint:
3x + 3y ≤ 15
Divide by 3:
x + y ≤ 5 (revised flour constraint)
Now we can graph these inequalities on a graph:
1. Plot the line x + y = 5.
2. Plot the line 2x + y = 8.
3. Shade the region below both lines.
4. The feasible region is the overlapping shaded area.
To find the optimal solution within this feasible region, we want to maximize the objective function.
Start by evaluating the objective function at the vertices (corner points) of the feasible region.
Calculate the objective function values when:
- (0, 0) - the origin
- (0, 5)
- (4, 0)
- The point where the two lines intersect.
Compare the objective function values at these points to identify the one that maximizes profit. This point represents the optimal solution to the problem.
Remember to verify that the constraints are satisfied at the optimal solution.
If you want to learn more about linear programming and how to solve it graphically or algebraically, here are some resources that could be helpful:
1. Khan Academy: Linear programming (https://www.khanacademy.org/math/ap-calculus-ab/ab-applications-of-definite-integrals-new/ab-8-4/a/linear-programming)
2. PurpleMath: Linear Programming (https://www.purplemath.com/modules/linprog.htm)
3. MIT OpenCourseWare - Linear Optimization (https://ocw.mit.edu/courses/sloan-school-of-management/15-053-optimization-methods-in-management-science-spring-2013/tutorials/MIT15_053S13_tut05.pdf)
These resources provide explanations, examples, and practice problems to help you better understand linear programming and how to solve it using different methods.
surely your text has a section on linear programming, where it explains problems just like this one.
Assign variables
write the constraints
define the objective function
evaluate at all the vertices of the region.