You cut square corners from a piece of cardboard that has dimensions 32 cm by 40 cm. You then fold the cardboard to create a box with no lid. To the nearest centimeter, what are the dimensions of the box that will have the greatest volume?

h = 6 cm

l = (32-2 (6)) = 20 cm

w = (40-2 (6)) = 28 cm

Step-by-step explanation:

Given in the question a piece of cardboard having dimensions 32 by 40

Suppose we cut square of length x

When you cut square corners from a piece of cardboard, two sides are 32-2x, and other two sides are 40-2x lengths

Step 1

Formula for the volume of box

v = height * length * width

v = x (32-2x) (40-2x)

v = 4x³ - 144x² + 1280x

Step 2

Find derivative and equal it to 0

dv/dx = 4 (3) x² - (2) 144x + 1280

0 = 12x² - 288x + 1280

x1 = 18.110

x2 = 5.88

Step 3

Select x and plug value of x to find dimension

(32-2x) > 0

2x < 32

x < 16

so domain say that x should be less than 16 which means x = 5.88 ≈ 6 cm

h = 6 cm

l = (32-2 (6)) = 20 cm

w = (40-2 (6)) = 28 cm