On a rectangular piece of cardboard with perimeter 13 inches

On a rectangular piece of cardboard with perimeter 13 inches, three parallel and equally spaced creases are made (see Figure 1). The cardboard is then folded along the creases to make a rectangular box with open ends (see Figure 2). Letting x represent the distance (in inches) between the creases, use the ALEKS graphing calculator to find the value of x that maximizes the volume enclosed by this box. Then give the maximum volume. Round your responses to two decimal places.

Solution

perimeter of cardboard = 13

length of rectangular box = x

width of rectangular box = x

height = (13 - 4x )/ 2

hence volume = length * width * height

V(x) = x^2 ( 13 - 4x) / 2

to find x that would maximize the volume find first derivative of V (x)

V\'(x) = -x ( 6x - 13 )

set the first derivative equal to 0 and solve for x

x = 13/6 = 2.16 inches

hence x = 2.16 would maximize the volume

maximum volume = 2.16^2 ( 13-4*2.16) / 2 = 10.22 in^3