Aimee is shipping a care package to her friend in college, and she has a piece of cardboard with which to create a box. The dimensions of the cardboard are 20 centimeters by 40 centimeters and she will create a box of height x

What value of x will provide the maximum volume of the box?

What is the maximum volume of the box?

x = 10 - 10sqrt (3) / 3

Approximately 4.2265 cm

Volume = 1539.6 cm³

Step-by-step explanation:

Let the height be x

V = (20 - 2x) (40 - 2x) x

V = (800 - 80x - 40x + 4x²) x

V = 4x³ - 120x² + 800x

dV/dx = 12x² - 240x + 800 = 0

3x² - 60x + 200 = 0

x = [60 + / - sqrt (60² - 4 (3) (200)) ]/6

x = [30 + / - 10sqrt (3) ]/3

x = 10 + / - 10sqrt (3) / 3

Approximately,

15.7735 and 4.2265

d²V/dx² = 24x - 240

For max, d²V/dx² < 0

So, x = 4.2265

V = (20 - 2x) (40 - 2x) x

Plug in x = 10 - 10sqrt (3) / 3

V = 1539.600718