SOLUTION: A rectangular bin with an open top and volume of 38.72 cubic feet is to be built. The length of its base must be twice the width and the bin must be at least 3 feet high. Material

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Question 1132560: A rectangular bin with an open top and volume of 38.72 cubic feet is to be built. The length of its base must be twice the width and the bin must be at least 3 feet high. Material for the base of the bin costs $12 per square foot. If it costs $538.56 to build the bin, what are its dimensions?
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!
Data:
width=x
lenght=2x
height=y%3E=3
Volume=length+%2A+width+%2A+height
Therefore:
%282x%29%28x%29%28y%29=38.72
2x%5E2%2Ay=38.72
Area of the base: length+%2Awidth
Area of the base=%282x%29%28x%29=2x%5E2
Cost of the base=12%282x%5E2%29=24x%5E2
Area of the sides=2%28length+%2A+height%29+%2B+2%28width+%2A+height%29
Area of the sides=2%282xy%29%2B2%28xy%29=4xy%2B2xy=6xy
Cost of the sides=8%286xy%29=48xy

Therefore:
Cost of the base + cost of the sides=538.56
=>24x%5E2%2B48xy=538.56
We have this system of equations:
2x%5E2y=38.72
24x%5E2%2B48xy=538.56
We can solve this system of equations by substitution method:
2x%5E2y=38.72
+y=38.72+%2F+2x%5E2
y=19.36+%2Fx%5E2
and
24x%5E2%2B48x%2819.36+%2F+x%5E2%29=538.56
24x%5E2%2B929.28%2Fx=538.56
24x%5E3%2B929.28=538.56x
I have solved this equation graphically and the result would be:
Solutions:
x ≈ 3.237
x ≈ 2.2
x ≈ -5.437-> disregard negative solution
check:
x ≈ 3.237
y=19.36+%2F3.237%5E2=1.84765
This value is not correct , because "y" would be < 3ft

And
x ≈ 2.2
y=19.36+%2F2.2%5E2=4
This value is correct, if "y" is at least 3 ft high.
Therefore:
width=x=2.2
lenght=2x=2%282.2%29=4.4
y=19.36+%2F+x%5E2=19.36+%2F+%282.2%29%5E2=4
Answer: it´s dimensions would be:
width=2.2ft
length=4.4ft
height=4ft