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Question 1151594: You make decorative stones for landscaping. A ton of coarse stones requires 2 hours of crushing, 5 hours of sifting, and 8 hours of drying. A ton of fine stones requires 6 hours of crushing, 3 hours of sifting, and 2 hours of drying. The coarse stones sell for $400 per ton. The fine stones sell for $500 per ton. In a work week your plant is capable of 36 hours of crushing, 30 hours of sifting, and 40 hours of drying.
Use corner solution, to Determine:
a. How much of each kind of stones you should make to maximize your revenue.
b. How much revenue you'll make at the maximum.
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! x = number of tons of coarse stones.
y = number of tons of fine stones.
make a table as shown below:
description coarse stones fine stones
number of tons x y
crushing 2 6 <= 36
sifting 5 3 <= 30
drying 8 2 <= 40
revenue 400 500 maximize
your objective function is:
400x + 500y
this is what your want to maximize.
your constraint functions are:
2x + 6y <= 36
5x + 3y <= 30
8x + 2y <= 40
x >= 0
y >= 0
using the decmos.com calculator, you would graph the opposite of the constraints
for example:
2x + 6y <= 36 is graphed as 2x + 6y >= 36
the area of the graph that is NOT shaded is the region of feasibility.
the maximum revenue will be at the corner points of this region.
the equations used for graphing are shown below
the graph is shown below:
the objective function is evaluated at each corner point.
the corner point with the maximum revenue is the solution.
(0,6) becomes 400 * 0 + 500 * 6 = 3000
(3,5) becomes 400 * 3 + 500 * 5 = 3700
etc.
the maximum revenue is found at (3,5).
all the constraints have to be satisfied.
this is true for (3,5) so that solution appears to be good.
some adjustment might need to be made at the point with fractions in it if only integer tons can be sold.
that would be (4.286, 2.857).
the options would be:
(4,2)
4,3)
(5,2)
(5,3)
the only one of those points where all the constraints were met was (4,2) and that point did not produce maximum revenue.
this analysis might be moot since (4.286,2.857) did not provide maximum reve nue, but it's still a consideration that might need to be explored.
based on the graphical analysis using desmos.com calculator, the maximum revenue is when 3 tons of coarse stones and 5 tons of fine stones are sold.
there is a simplex method tool that can also be used.
graphical solutionis only good for 2 variables.
simplex method tool can be used for more than 2 variables.
this tool can be found at:
https://www.zweigmedia.com/RealWorld/simplex.html
i used this tool to see if it came up with the same solution as the graphical solution.
it did.
using this tool, you provide the constraints as is and you do not provide the opposite of the constraint inequalities.
the results are shown below:
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