A candy company makes a chocolate nut mix and and a chocolate fruit mix. The nut mix uses 2 pounds of chocolate, 2 pounds of nuts and no fruit. The fruit mix uses 4 pounds of chocolate,no nuts and 2 pounds of fruit. The nut mix brings in $7 profit and the fruit mix brings in $8 profit. If the company has 1000 pounds of chocolate, 900 pounds of nuts, and 400 pounds of fruit in stock, how many boxes of each mix should it make to maximize profit? What is that profit?
If there are x boxes of nuts and y boxes of fruit, you want to
maximize p=7x+8y subject to
2x+4y <= 1000
2x <= 900
2y <= 400
Now use your favorite linear programming tool.
To maximize the profit, we need to determine how many boxes of each mix should be made within the given constraints.
Let's start by defining the variables:
- Let 'x' represent the number of boxes of the chocolate nut mix.
- Let 'y' represent the number of boxes of the chocolate fruit mix.
Now, let's set up the equations representing the constraints and the objective function:
1. Constraint for available chocolate:
The nut mix uses 2 pounds of chocolate per box, and the fruit mix uses 4 pounds of chocolate per box. The total available chocolate is 1000 pounds, so the equation is:
2x + 4y ≤ 1000
2. Constraint for available nuts:
The nut mix uses 2 pounds of nuts per box, and the fruit mix doesn't use any nuts. The total available nuts are 900 pounds, so the equation is:
2x ≤ 900
3. Constraint for available fruit:
The nut mix doesn't use any fruit, and the fruit mix uses 2 pounds of fruit per box. The total available fruit is 400 pounds, so the equation is:
2y ≤ 400
4. Non-negativity constraint:
We cannot have negative numbers of boxes, so x ≥ 0 and y ≥ 0.
Now, let's define the objective function, which is the total profit:
Total Profit = 7x + 8y
We need to maximize this objective function.
Now, we can solve this linear programming problem using a graphical method or the simplex method. Let's use the graphical method to find the optimal solutions.
First, let's graph the feasible region for the constraints:
Graph:
Draw the lines representing the inequalities and shade the region that satisfies all the constraints.
After graphing the feasible region, we can then proceed to find the corner points (vertices) of the region.
The corner points of the feasible region are the intersections of the lines. We can find these points by solving the equations pairwise. In this case, we have three vertices:
A: (0, 200)
B: (300, 0)
C: (350, 25)
Now, we can evaluate the total profit at each of these corner points:
- At point A: Total Profit = 7(0) + 8(200) = 1600
- At point B: Total Profit = 7(300) + 8(0) = 2100
- At point C: Total Profit = 7(350) + 8(25) = 2625
To maximize the profit, we choose the corner point with the highest profit, which in this case is point C (350, 25).
Therefore, the company should make 350 boxes of the chocolate nut mix and 25 boxes of the chocolate fruit mix to maximize profit.
The maximum profit would be $2625.