Four small squares of side x are cut out of a square of side 12 cm to make a tray by folding the edges. What is the value of so that the tray has the maximum volume?
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When small square of side x is cut at the corners, the dimensions will reduce by 2x
Area of base = (12 – 2x)(12 – 2x)
= 144 – 48x + 4x^2
Height of tray = x
Volume = base area x height
V = (144 – 48x + 4x^2)x
= 144x – 48x^2 + 4x^3
At maximum volume:
dV/dx = 0
144 – 96x + 12x^2 = 0
12x^2 – 96x + 144 = 0
x^2 – 8x + 12 = 0
x^2 – 6x – 2x + 12 = 0
x(x – 6) -2(x – 6) = 0
(x – 2)(x – 6) = 0
Either x = 2 or x = 6
Area of base = (12 – 2x)(12 – 2x)
= 144 – 48x + 4x^2
Height of tray = x
Volume = base area x height
V = (144 – 48x + 4x^2)x
= 144x – 48x^2 + 4x^3
At maximum volume:
dV/dx = 0
144 – 96x + 12x^2 = 0
12x^2 – 96x + 144 = 0
x^2 – 8x + 12 = 0
x^2 – 6x – 2x + 12 = 0
x(x – 6) -2(x – 6) = 0
(x – 2)(x – 6) = 0
Either x = 2 or x = 6
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