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Christopher is making necklace and key chains.
He can make a necklace in 0.5 hours and key chain in 0.25 hours and he has no more than 20 hours.
The cost in making a necklace is $2 while for key chain it is $3. He could invest up-to $120.
Write a system of four linear inequalities and find how many of each he could make?
If he sells each necklace for $10 and key chain for $8, tell when he will get maximum revenue and maximum profit?
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Let X be the number of necklaces and Y be the number of key chains.
Then the inequalities are
X >= 0;
Y >= 0;
0.5X + 0.25Y <= 20 (the restriction by time);
2X + 3Y <= 120 (the restriction by investing)
The inequalities describe and represent this quadrilateral in the first quadrant,
restricted by the x- and y-axes and the red and the green lines:
Plots y = (red) and y = (green)
The vertices of this quadrilateral are
P1 = (0,0) = the origin of the coordinate system,
P2 = (0,60) = the y-intersect of the green line,
P3 = (40,0) = the x-intersect of the red line, and
P4 = (30,20) = the intersection point of the red and the green lines.
(this intersection point is the solution of the system 0.5X + 0.25Y = 20 and 2X + 3Y = 120).
The solution for the maximum profit
The profit function is P(X,Y) = ($10 - $2)*X + ($8 - $3)Y = 8X + 5Y dollars. (1)
To find the maximum profit, you should check the values of the profit function P(x,Y) at all four vertices and then compare four values.
The values are:
at P1: P(X,Y) = P(0,0) = 8*0 + 5*0 = 0;
at P2: P(X,Y) = P(0,60) = 8*0 + 5*60 = 300;
at P3: P(X,Y) = P(40,0) = 8*40 + 5*0 = 320;
at P4: P(X,Y) = P(30,20) = 8*30 + 5*20 = 340.
The maximal value of the profit function is at P4, and it gives the optimal solution:
The maximum profit is $340 and it is achieved when Christoper produces 30 necklaces and 20 key chains.
The solution for the maximum revenue
The revenue function is R(X,Y) = $10*X + $8*Y = 10X + 8Y dollars. (2)
To find the maximum revenue, you should check the values of the revenue function R(x,Y) (2) at all four vertices and then compare four values.
The values are:
at P1: R(X,Y) = R(0,0) = 10*0 + 8*0 = 0;
at P2: R(X,Y) = R(0,60) = 10*0 + 8*60 = 480;
at P3: R(X,Y) = R(40,0) = 10*40 + 8*0 = 400;
at P4: R(X,Y) = R(30,20) = 10*30 + 8*20 = 460.
The maximal value of the revenue function is at P2, and it gives the optimal solution for the revenue:
The maximum revenue is $480 and it is achieved when Christoper produces 0 necklaces and 60 key chains.
As you see, maximum profit and maximum revenue are achieved at different points and assume different strategies.
The method I solved this problem is called "the linear programming method".
It may have different / (other) names, too.