If 10,800 cm2 of material is available to make a box with a square base and an o

Question

If 10,800 cm2 of material is available to make a box with a square base and an open top, find the largest possible volume of the box.

Explanation / Answer

1) Let each side of the base be = x cm; and its height = y cm 2) ==> Surface Area(SA) = base area + four sides area [since its open at top] ==> SA = x^2 + 2y(x + x) = x^2 + 4xy = 10800 cm^2 ==> y = {10800 - x^2}/4x = 2700/x - x/4 cm 3) Volume (V) = (x^2)*(2700/x - x/4) = 2700x - (x^3)/4 4) Differentiating, V' = 2700 - (3/4)*(x^2) 5) Setting V' = 0, 2700 - (3/4)*(x^2) = 0; Solving x = +/- 60 cm As dimension cannot be negative, -60 is rejected. As well V" = (-3/2)*(x) At x = -60, V" = 90, which is > 0; so volume is minimum; while at x = 60, V" = -90, which is < 0; so volume is maximum. Hence for maximum volume x = 60 cm.; ==> height y = 30 cm Hence greatest volume possible = 60 x 60 x 30 = 1,08,000 cm^3

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