Answer:
(25 -5√7)/3 ≈ 3.924 inches
Explanation:
After removing squares from the corners and folding up the sides, the open-top box will have dimensions ...
x × (20 -2x) × (30 -2x)
The maximum value for this cubic expression in the domain 0 < x < 10 is found easily by a graphing calculator to be at x = 3.924 inches.
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You can solve this by expanding the cubic and finding the solution that makes its derivative equal to 0.
V(x) = 4x^3 -100x^2 +600x
The derivative is ...
V'(x) = 12x^2 -200x +600 = 0
Dividing by 12, we have ...
x^2 -50/3x +50 = 0
Completing the square, we get the form ...
(x -25/3)^2 = -50 +(25/3)^2 = (-450 +625)/9 = 175/9
Taking the square root gives ...
x -25/3 = ±(5√7)/3
The solution in the desired range is ...
x = (25 -5√7)/3 ≈ 3.924 . . . inches
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Additional comment
We limit the value of x to half the width of the cardboard material, as any longer cuts would overlap to create a negative box dimension. The graph shows the maximum volume to be about 1056.3 cubic inches. Box dimensions would be 3.924 in deep by 12.153 in by 22.153 in.
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The generic solution can be found using a similar procedure. For cardboard dimensions a×b, it will be ...
x = ((a +b) -√((a -b)² +ab))/6