# calculus

an open box is to be made from a piece of metal 16 by 30 inches by cutting out squares of equal size from the corners and bending up the sides. what size should be cut out to create a box with the greatest volume? what is the maximum volume?

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1. Let each sides of the squares to be cut out be x inches
then base of box = 30-2x
width of box = 16-2x
height of box = x

V = x(16-2x)(30-2x)
= 480x - 92x^2 + 4x^3
dV/dx = 480 - 184x + 12x^2
= 0 for a max of V

12x^2 - 184x + 480 = 0
3x^2 - 46x + 120 = 0
(x-12)(3x - 10)
x = 12 or x = 10/3
but clearly 0 < x < 8 , for x=12 the width would be negative

so x = 10/3

Max Volume = (10/3)(16-20/3)(30-20/3) = appr. 725.9 inches^3

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