Find the maximum volume in the given question
Answers
The height of box is (4 - √7)/3 = 0.45 m such that the volume of the box is maximum
Step-by-step explanation:
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 = (6 - 2x)(2 - 2x)x
= 2(3 - x)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)/(2 * 3)
= ( 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