SOLUTION: 1. A book seen below is to have a square base, an open top and volume 32 cubic units. if x is the length of each side of its bases and y is its height, how many units should x and

Question 1065987: 1. A book seen below is to have a square base, an open top and volume 32 cubic units. if x is the length of each side of its bases and y is its height, how many units should x and y be in order to make the box with the smallest amount of materials?
2.
Found 2 solutions by Boreal, rothauserc:
Answer by Boreal(15235) (Show Source): You can put this solution on YOUR website!
The base is x^2 and the height y. It has a volume=32, so 32=x^2y
The area of the box is x^2 (bottom) + 4xy (the sides).
Therefore, the area is x^2+4x*32/x^2 by substitution.
Take the derivative and set it equal to 0.
x^2+4*32/x is the function; the derivative is 2x-128/x^2=0
x=64/x^2
x^3=64
x=4 units
y=2 units
area is 16+4*8=48 square units
Can try x=5 and y=1.3. This gives a volume of 32.25 cubic units but area is 25+26=51 square units.
Can try x=2 and y=8. This gives a volume of 32 cubic units but area is 4+64=68.

Answer by rothauserc(4718) (Show Source): You can put this solution on YOUR website!
We are given
:
1) x^2 * y = 32
:
2) x^2 + 4xy = surface area(SA)
:
we want SA to be the minimum
:
solve for y in equation equation 1)
:
y = 32 / x^2
:
now substitute for y in equation 2)
:
A = x^2 + 4x(32/x^2) = x^2 + (128/x)
:
take the first derivative
:
A' = 2(x^3 - 64) / x^2
:
set first derivative = 0 and solve for x
:
2(x^3 - 64) / x^2 = 0
:
2(x^3 - 64) = 0
:
x^3 = 64
:
x = 4
:
y = 32 / 4^2 = 2
:
SA = 48 square units
:
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x = 4 units, y = 2 units
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:

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