Algebra 2 - Multiple Questions

Question

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?



Answer 1

As Paul M. points out in his comment, we can't give you the answer because you didn't provide the diagram. However, here is the procedure. You should be able to follow it to get the answer for yourself:

a) The shaded region is called the "feasible region". It is the set of all possible points that are solutions to the objective function given the constraints.

b) The vertices are the points where the various lines that form the feasible region meet. Locate them on your diagram, label them (A, B, C, etc.), then write down their coordinates (A = (1,3), B = (5, 3), etc.)

c) The objective function is the profit the farmer makes from selling berries: $3 for each box of blueberries (x) and $5 for each box of strawberries (y)

P = $3x + $5y

d) The optimal value occurs at one of the vertices. Plug in the (x,y) coordinates of each vertex into the profit equation (objective function) above and compute the profit. The largest value of P you get is the optimal profit.

e) The (x,y) coordinates of the vertex that yields the optimal profit identify the number of blueberries (x) and strawberries (y) that the farmer should buy.