determine the dimensions of the box that can be constructed at minimum cost.

Calculus
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Jul 16th, 2015

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A rectangular box is to have a square base and a volume of 20 feet cubed. If the material for the base costs $.30/square foot, the material for the sides costs $.10/square foot, and the material for the top costs $.20/square foot, determine the dimensions of the box that can be constructed at minimum cost
:
Let y = the cost of the box
:
Let x = one side of the square base
:
Let h = height of the box
:
We know that the box has to be 20 cu ft:
x^2*h = 20
h = (20/x^2)
:
Costs:
Base = .30(x^2)
Top = .20(x^2)
1 side = .10(x*(20/x^2)) = .10(20/x)
4 sides = .40(20/x) = 8/x 
:


Total cost:
y = Base + Top + 4 sides 
y =.30(x^2) + .20(x^2) + 8/x
y = .5x^2 + 8/x
Graph it:
+graph%28+300%2C+200%2C+-6%2C+6%2C+-6%2C+12%2C+.5x%5E2+%2B+%288%2Fx%29%29+
:
Looks like min cost (y) occurs when x = 2 ft, looks like a cost(y) of $6.00
:
Then the height = 20/2^2 = 5 ft
:
What is the cost then?
top .30(2^2) = 1.20
bot .20(2^2) = 0.80
hth .10(2*5) = 1.00 * 4 sides = $4.00
Total cost = $6.00 like the graph indicates.

:
Check: 2 * 2 * 5 = 20 cu ft as required 

x = side of square base in feet
h = height of box in feet
cost of base = [img src="http://www.algebra.com/cgi-bin/plot-formula.mpl?expression=.30x%5E2&x=0003" align="MIDDLE" alt=".30x%5E2" >
cost of 4 sides = .10%2A4%2Ah%2Ax
cost of tgop = .20x%5E2
C+=+.3x%5E2+%2B+.4xh+%2B+.2x%5E2
%28x%5E2%29h+=+20 volume of box = 20 ft3
h+=+20%2Fx%5E2
C+=+.3x%5E2+%2B+.4x%2A%2820%2Fx%5E2%29+%2B+.2x%5E2
C+=+.3x%5E2+%2B+8%2Fx+%2B+.2x%5E2
C+=+.5x%5E2+%2B+8%2Fx
I make a table
x = 1
C = 8.5
-------
x = 2
C = 6
-------


x = 3
C = 7.17
-------
x = 4
C = 10
-------
It looks like x = 2 may give a minimum cost
I tried x = 1.8 and x = 2.2
-------
x = 1.8
C = 6.06
-------
x = 2.2
C = 6.06
-------
so, C=6 really is a min
x%5E2h+=+20
4h+=+20
h+=+5
2x2x5 are the dimensions for min cost = $6.00

Please let me know if you nea case for torture by michale levined any clarification. I'm always happy to answer your questions.
Jul 16th, 2015

i am vert confused as to what the answers are 

Jul 16th, 2015

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