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