Question

A box with an open top is to be constructed from a square piece ofcardboard, with sides 6 meters in length, by cutting a square from each of the fourcorners and bending up the sides. Find the dimensions that maximize the volume ofthe box and the maximum volume.

Answer 1

Explanation:

Let x be the height of the box . x will be cut at four corners .

each side of the box = 6 - 2x

volume of box V = ( 6 - 2 x )² x

V = ( 36 + 4 x² - 24 x ) x

V = 4x³ - 24 x ² + 36 x

For maximum volume

dV / dx = 12 x² - 48 x + 36 = 0

x² - 4 x + 3 = 0

( x - 3 ) ( x - 1 ) = 0

x = 1 , x = 3 ( not possible )

Possible solution x = 1

Volume , for x = 1

V = V = ( 6 - 2 x )² x

V = 4² x 1

= 16 m³.