Question 1088781: The Nut Factory produces a mixture of peanuts and cashews. The company guarantees that at least 40% of the total weight is cashews. It has a contract to produce 1000 pounds or more of the mixture. The peanuts cost $0.80 per pound, and the cashews cost $1.50 per pound.
(a) Find the amount of each kind of nut the company should use to minimize the cost if 870 pounds of peanuts are available.
peanuts lb
cashews lb
cost $
(b) Find the amount of each kind of nut the company should use to minimize the cost if 390 pounds of peanuts are available.
peanuts lb
cashews lb
cost $
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! use of the desmos.com calculator allows you to enter the equations and inequalities as is rather than having to solve for y.
that calculator can be found here:
https://www.desmos.com/calculator
when you use this calculator, the easiest and clearest way to find the region of feasibility is to find the area of the graph that is NOT shaded.
in order to do this, you enter your inequalities opposite of what they are.
you'll see what i mean as i work through your problem.
let x equal the number of pounds of peanuts.
let y equal the number of pounds of cashews.
the total cost is .80 * x + 1.50 * y
that's because the peanuts cost 80 cents a pound and the cashews cost 1.50 a pound.
this is what you want to minimize and is therefore your objective function.
you have a contract to deliver at least 1000 pounds of the mixture.
x + y >= 1000.
the number of pounds of peanuts or cashews must be greater than or equal to 0.
x >= 0
y >= 0
the company guarantees that at least 40% of the mixture will be cashews.
y >= .4 * (x + y)
to summarize:
your objective function is .8 * x + 1.5 * y which is what you want to minimize.
your constraints are:
x + y >= 1000
x >= 0
y >= 0
y >= .4 * (x + y)
you graph the opposite of your constraints.
you graph:
x + y <= 1000
x <= 0
y <= 0
y <= .4 * (x + y)
for the first analysis, the number of peanuts must be less than or equal to 870 pounds.
x <= 870
you would graph x >= 870.
for the second analysis, the number of peanuts must be less than or equal to 390 pounds.
you would graph x >= 390.
so, for the first analysis, your constraints are:
x + y >= 1000
x >= 0
y >= 0
y >= .4 * (x + y)
x <= 870
and you would graph:
x + y <= 1000
x <= 0
y <= 0
y <= .4 * (x + y)
x >= 870
for the second analysis, your constraints are:
x + y >= 1000
x >= 0
y >= 0
y >= .4 * (x + y)
x <= 390
and you would graph:
x + y <= 1000
x <= 0
y <= 0
y <= .4 * (x + y)
x >= 390
for the first analysis, your graph would look like this:
the corner points of the feasible region are:
(0,1000)
(600,400)
(870,580)
you analyze the objective function at each of the corner points.
the corner point with the smallest value is your solution.
the objective function is .8 * x + 1.5 * y
the cost at (0,1000) = .8 * 0 + 1.5 * 1000 = 1500
the cost at (600,400) = .8 * 600 + 1.5 * 400 = 1080
the cost at (870,580) = .8 * 870 + 1.5 * 580 = 1566
the minimum cost is at (600,400)
that's 600 pounds of peanuts and 400 pounds of cashews.
all the constraints are satisfied.
600 pounds of peanuts is less than or equal to 870
1000 pounds of peanuts plus cashews is greater than or equal to 1000
pounds of peanuts and cashews are each greater than or equal to 0.
400 pounds of cashews / 1000 = 40% which is greater than or equal to 40%.
for the second analysis, your graph would like this:
the corner points of the feasible region are:
(0,1000)
(390,610)
you analyze the objective function at each of these corner points.
the cost at (0,1000) = .8 * 0 + 1.5 * 1000 = 1500
the cost at (390,610) = .8 * 390 + 1.5 * 610 = 1227
your minimum cost is at (390,610)
that's 390 pounds of peanuts and 610 pounds of cashews.
all the constraints are satisfied.
1000 pounds of peanuts and cashews is greater than or equal to 1000.
pounds of peanuts and cashews are each greater than or equal to 0.
610 pounds of cashews / 1000 = .61 = 61% which is greater than or equal to 40%.
the answer to each of your questions is shown below:
(a) Find the amount of each kind of nut the company should use to minimize the cost if 870 pounds of peanuts are available.
peanuts lb = 600
cashews lb = 400
cost $ = 1080
(b) Find the amount of each kind of nut the company should use to minimize the cost if 390 pounds of peanuts are available.
peanuts lb = 390
cashews lb = 610
cost $ = 1227
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