Question

I NEED HELP PLEASE, THANKS! :)

The senior class is having a fundraiser to help pay for the senior trip. Selling a box of chocolates yields a profit of $2.45, while selling a box of cookies yields a profit of $2.70. The demand for cookies is at least twice that of chocolates, but the amount of cookies produced must be no more than 550 boxes plus 3 times the number of chocolates produced. Assuming that the senior class can sell every box that they order, how many boxes of each should they order to maximize profit if they cannot order more than 1950 boxes combined? (Show work)

Answer 1

1600 boxes of cookies; 350 boxes of chocolates

Explanation:

Let x and y represent the numbers of boxes of cookies and chocolates to order, respectively. The constraints seem to be ...

• x ≥ 2y
• x ≤ 550 +3y
• x + y ≤ 1950

The objective function we want to maximize is ...

p = 2.70x +2.45y

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Looking at the problem, we see that cookies yield the most profit, so we'd like to maximize the boxes of cookies sold within the allowed limits.

There are two limits on x:

x ≤ 550 +3y

x ≤ 1950 -y

x will be as large as it can be if it bumps up against these limits simultaneously:

550 +3y = 1950 -y

4y = 1400 . . . . . . . . . add y-550

y = 350

x = 1950 -350 = 1600

Profit will be maximized for an order of 1600 boxes of cookies and 350 boxes of chocolates.

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The "feasible region" of the solution is where the three constraint inequalities overlap. It is a quadrilateral with vertices at (0, 0), (550, 0), (1300, 650), and (1600, 350). We can reject the ones with x or y being 0. Per our analysis above, the solution of interest is (x, y) = (1600, 350), since it maximizes x.

The usual protocol for solving a linear programming model like this is to evaluate the objective function at each of the vertices. With a little thought about the problem, we have saved some evaluation effort.