SOLUTION: Fruit-for-Africa (Pty) Ltd produces two gift packages of fruits. Package A contains 20 peaches,15 apples and 10 pears. Package B contains 10 peaches, 30 apples and 12 pears. Fruit-

Algebra ->  College  -> Linear Algebra -> SOLUTION: Fruit-for-Africa (Pty) Ltd produces two gift packages of fruits. Package A contains 20 peaches,15 apples and 10 pears. Package B contains 10 peaches, 30 apples and 12 pears. Fruit-      Log On


   

Question 1183897: Fruit-for-Africa (Pty) Ltd produces two gift packages of fruits. Package A contains 20 peaches,15 apples and 10 pears. Package B contains 10 peaches, 30 apples and 12 pears. Fruit-for-Africa has 40 000 peaches, 60 000 apples and 27 000 pears available for packaging. The profit on package A is R2.00 and profit on B is R2.50. Assuming that all fruit packed can be sold, what number of packages of type A and B should be prepared to maximize the profit?
i. Write down the linear programming problem
ii. Graphically depict the constraints and shade the feasible region
Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
x = number of package A's.
y = number of package B's.

your constraint inequalities are.

20x + 10y <= 40000
15x + 30y <= 60000
10x + 12y <= 27000

your objective function is:

profit = 2x + 2.5y

using the desmos.com calculator, you would graph the opposite of these inequalities.
this calculator can be found at https://www.desmos.com/calculator

specifically, your would graph....

20x + 10y >= 40000
15x + 30y >= 60000
10x +12y >= 27000

the area of the graph that is not shaded is your region of feasibility.
the corner points of this region qare where your maximum profit will lie.
you evaluate each corner point to find the corner point with the maximum profit.

the graph can be shown below:



the corner point with the maximum profit is at (750,1625).
this means the value of x is 750 and the value of y is 1635.
the objective function ia 2 * 750 + 2.5 * 1625 = 5562.5
that is the maximum profit.

all the constraints need to be satisfied.
at (750,16250) the constraints are:

750 * 20 + 1625 * 10 = 31250 which is <= 40000
750 * 15 + 30 * 1625 = 60000 which is <= 60000
750 * 10 + 1625 * 12 = 27000 which is <= 27000

you also has the constraint the x and y both need to be greater than 0, which has also been satisfied.