SOLUTION: An open display box is to be made in the shape of a rectangular prism is to be made from a 3ft x 8ft piece of lumber by cutting equal squares from each of the four corners and turn
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Question 258993: An open display box is to be made in the shape of a rectangular prism is to be made from a 3ft x 8ft piece of lumber by cutting equal squares from each of the four corners and turning up the sides. Find the volume of the largest box that can be made.
From reading the question, I know it is an optimization problem. I know that the volume formula of a rectangle is length x width x height. Then I get confused.
Answer by stanbon(75887) (Show Source): You can put this solution on YOUR website!
An open display box is to be made in the shape of a rectangular prism is to be made from a 3ft x 8ft piece of lumber by cutting equal squares from each of the four corners and turning up the sides. Find the volume of the largest box that can be made.
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Draw a picture of the 3' by 8' rectangular piece.
Sketch a square in each of the 4 corners that is x by x.
Imagine cutting out the squares and folding up the sides.
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Volume = (base)(height)
Base = (3-2x)(8-2x)
height = x
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Volume = (24 -6x -16x + 4x^2)(x)
V(x) = 4x^3 - 22x^2 + 24x
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To find the maximum determine the derivative:
V'(x) = 12x^2 - 44x + 24
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Solve 12x^2 - 44x + 24 = 0
3x^2 - 11x + 6 = 0
3x^2-9x-2x+6 = 0
3x(x-3)-2(x-3) = 0
(x-3)(3x-2) = 0
x = 3 or x = 2/3
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If x = 3 the volume is zero because the dimension (a minimum)
If x = 2/3 the volume is a maximum.
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V(x) = 4x^3 - 22x^2 + 24x
V(2/3) = 7.4074 cu ft.
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Cheers,
Stan H.
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