A farmers’ market manager needs to purchase pints of blueberries and pints of strawberries. His display for berries can hold no more than 50 pints. His customers by no more than 25 pints of blueberries and no more than 40 pints of strawberries. He is able to sell blueberries for $3 per pint and strawberries for $5 per pint. Below is the graph of constraints. Let X represent blueberries and Y represent strawberries.
0. What is the name of the shaded region in the graph?
0. What are the vertices?
0. What is the object function?
0. What is the optimal value of profit?
0. How many pints of blueberries and strawberries should the manager purchase in order to optimize his profit?
Also, I'm not sure how to solve, so if you could go through this step by step that would be great. Thank you :)
Okay, thank you so much
To answer these questions, we need to analyze the given information and solve a linear programming problem.
1. The shaded region in the graph represents the feasible region. It represents all the possible combinations of pints of blueberries and strawberries that satisfy the given constraints.
2. The vertices of the shaded region are the points at which the boundary lines intersect. These vertices are the corner points of the feasible region and define different combinations of blueberries and strawberries. By evaluating the objective function at each vertex, we can determine the optimal solution.
3. The objective function represents the quantity we want to maximize or minimize. In this case, the objective is to maximize profit. Since profit is calculated as the selling price multiplied by the quantity sold, the objective function can be defined as:
Profit = 3X + 5Y
Here, X represents the number of pints of blueberries, and Y represents the number of pints of strawberries.
4. To find the optimal value of profit, we need to evaluate the objective function at each vertex and compare the results. The vertex that gives the maximum value will correspond to the optimal profit. In other words, we need to plug in the values of X and Y for each vertex into the profit function and find the maximum value.
5. To determine the optimal combination of pints of blueberries and strawberries, we need to solve the linear programming problem. We can do this by evaluating the objective function at each vertex and comparing the values.
However, without the graph or specific values for the vertices, it is not possible to directly answer questions 2, 4, and 5. These questions require calculations using the specific values of the vertices. Once you have the values for the vertices, you can plug them into the objective function to find the optimal profit and the corresponding quantities of blueberries and strawberries that should be purchased.
he wants to maximize revenue (or profit, since his costs are apparently zero).
So, if he sells b pints of blueberries
and s pints of strawberries
he wants to maximize
p = 3b+5s
subject to the constraints
b+s <= 50
0 <= b <= 25
0 <= s <= 40
I'm sure you can graph those lines. Now just find the points where they intersect, and those are the vertices.