A rectangular box is created from a piece of posterboard which is 24'' long and 9'' wide by cutting out identical squares from the corners and then turning up the sides Find the dimensions of the box?

Answer 1

First we need to find the side length of the square we cut in order to maximize the volume of the box. Let the side of the square be x.

Volume = lwh

l = 24 - 2x

w = 9 - 2x

h = x

V = (24 - 2x)(9 - 2x)(x)

V = [(24)(9)(x) - (24)(2x)(x) - (2x)(9)(x) - (2x)(-2x)(x)]

V = 216x - 48x^2 - 18x^2 + 4x^3

V = 216x - 60 x^2 + 4x^3 Take the derivative

V' = 216 - 120x + 12x^2 Make the derivative equal to 0

0 = 216 - 120x + 12x^2

0 = 12(18 - 10x + x^2)

0 = 18 - 10x + x^2

x = [10 ± √[(10^2 - 4(1)(18)]]/2

x = (10 ± √28)/2

x = (10 ± 2√7)/2

x = 5 ± √7 = x = 5 ± 2.6

x = 7.6 or x = 2.4 this are critical values, which we substitute into volume equation

V = 216(2.4) - 60(2.4^2) + 4(2.4^3) = 228 in^3

Thus, the side length of the square we cut is 2.4 in, which also is the height of the box. So,

l = 12 - 2(2.4) = 12 - 4.8 = 7.2

w = 9 - 2(2.4) = 9 - 4.8 = 4.2

Or we can estimate and say that the box will be 7 inches long, 4 inches wide, and 2 inches high.

Answer 2

231