# Optimization

Submitted by Timothy Ornelas on March 16, 2015 - 10:19pm

An open box is to be made out of a 10-inch by 18-inch piece of cardboard by cutting out squares of equal size from the four corners and bending up the sides. Find the dimensions of the resulting box that has the largest volume.

I need the dimensions of the bottom of the box and the height of the box.

Please help

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## Re: Optimization

Length = 18 - 2x

Width = 10 - 2x

Height = x

Volume = Length × Width × Height

$V = (18 - 2x)(10 - 2x)(x)$

$V = 180x - 56x^2 - 4x^3$

$\dfrac{dV}{dx} = 180 - 112x - 12x^2 = 0$

$x = 1.3978 ~ \text{ and } ~ -10.7311$

Use x = 1.4 inches

Dimensions of the largest box:

Length = 18 - 2(1.4) = 15.2 inches

Width = 10 - 2(1.4) = 7.2 inches

Height = 1.4 inches

Note: I did not re-check my solution.

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