From a rectangular sheet of metal of dimensions 8cm×5cm, equal square are cut off from the corners and the flaps are then folded upto form an open rectangular box.find tge side of the square cut so that the box may be of greater capacity.what is the maximum capacity of the box so made
Answers
Given : a rectangular sheet of metal of dimensions 8cm×5cm, equal square are cut off from the corners and the flaps are then folded upto form an open rectangular box
To find : the side of the square cut so that the box may be of greater capacity and capacity
Solution:
Let say side of square cut from corners = x
Then Size of Box = 8 - 2x , 5 - 2x , x
Volume of box = (8 - 2x)(5 - 2x)x
=> V = ( 4x² - 26x + 40)x
=> V = 4x³ - 26x² + 40x
dV/dx = 12x² - 52x + 40
dV/dx = 0
=> 12x² - 52x + 40 = 0
=> 3x² - 13x + 10 = 0
=> 3x² - 3x - 10x + 10 = 0
=> 3x(x - 1) - 10(x - 1) = 0
=> (3x - 10) (x - 1 ) = 0
=> x = 10/3 , 1
x = 10/3 not possible as then 5 - 2x will become - ve ( not possible )
d²V/dx² = 24x - 52
x = 1 => d²V/dx² = -28 < 0
=> Volume is maximum when x = 1
the side of the square cut so that the box may be of greater capacity = 1 cm
V = 4x³ - 26x² + 40x = 4 - 26 + 40 = 18
maximum capacity of the box = 18 cm³
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