Side Cutting
A rectangular piece of cardboard of side 24 by 36 inches is used to make an open-top box by cutting out a sma
A rectangular piece of cardboard of side 24 by 36 inches is used to make an open-top box by cutting out a small square from each corner and bending up the sides. In order for the box to have maximum volume, the edge of each of the cutouts should have length ANSWER inches.
The dimensions of the resulting box are x by 24 - 2x by 36 - 2x (helps to draw a picture). So the volume is 4x^3 - 120x^2 + 864x. The maximum occurs at a critical point, so we need roots of V' = 12x^2 - 240x + 864 = 0. Well, x^2 - 20x + 72 = 0 has roots 10 +/- sqrt(28) ~ 15.29, 4.71. The 15.29 solution would require subtracting 30.58 from the 24 inch side, which is impossible, leaving a 4.71 Inch Square from each side maximizing the volume. (You should verify that x = 4.71 is a local max by the first or second derivative test, but it's clear from the context that it has to be.)
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