Question

What is the maximum volume in cubic inches of an open box to be made from a 16-inch by 30-inch piece of cardboard by cutting out squares of equal sides from the four corners and bending up the sides?

Answer 1

let x be length of side of each cut -out square

V = x(16 - 2x)(30 - 2x) = x(480 - 92x + 4x^2) = 4x^3 - 92x^2 + 480x

dV/dx = 12x^2 - 184x + 480 = 0 for turning points

x = 12, 3 .333...

second derivative = 24x - 184

when x = 12 this is positive and when x = 3.33... its negative so we have a maximum when x = 3.33...

Maximum volume is 3.3333..( 16 - 2(3.33...)(30 - 3.33....)

= 725.92 in^3