Question

A rectangular sheet of metal of length 6 meters and width 2 meters is given. Four equal
squares are removed from the corners. The size of this sheet are now turned up to form
an open rectangular box. Find approximately, the height of the box, such that the
volume of the box is maximum

Answer 1

A rectangular sheet of metal of length 6m and width 2 m is given

Four equal squares are removed from the corners

Let say Square size = x * x m

Length of open rectangular box = 6 - 2x m width of open rectangular box = 2- 2x m

2-2x > 0

=> x < 1

Height = x m

Volume ( 62x)(2-2x)x

= 2(3x)2(1-x)x

=4x(3 + x² - 4x)

= 4x3 - 16x² + 12x

V = 4x³-16x² + 12x

dv/dx = 12x² - 32x + 12

12x²-32x + 12 = 0

=> 3x² - 8x + 3 = 0

x = (8 ± √(64-36) )/2(3)

= (8+2√7)/(23)

= (4 ± √7)/3

as x < 1

=> x = (4-√7)/3 = 0.45

d²V/dx² = 24x - 32 is - ve as x <1 Hence Volume is maximum

The height of box is (4-√7)/3 = 0.45 m such that the volume of the box is maximum

Learn more:

An open box is to be made from a square piece of material, 24 cm on