a storage tank is to be made from a metal sheet of metal 9 meters square by cutting a square side x meters from each corner and bending up the sides .what is the maximum volume of the tank if the height of the order ank must not exceed 3 meters
Answers
Answer: a storage tank is to be made from a metal sheet of metal 9 meters square by cutting a square side x meters from each corner and bending up the sides
To Find : what is the maximum volume of the tank
Solution:
Assuming open Tank :
metal sheet of metal 9 meters square
Hence side of sheet metal = 3 m
let say square cut form each side = x m
Then Base would be (3 - 2x ) m
Height would be x m
Volume = ( 3 - 2x)²x
=> V = (9 + 4x² - 12x)x
=> V = 4x³ - 12x² + 9x
dV/dx = 12x² - 24x + 9
dV/dx = 0
=> 12x² - 24x + 9 = 0
=> 4x² - 8x + 3 = 0
=> 4x² - 6x - 2x + 3 = 0
=> 2x( 2x - 3) - 1(2x - 3) = 0
=> (2x - 1)(2x - 3) = 0
=> x = 1/2 x = 3/2
x = 3/2 not possible as then 3 - 2x = 0
so x = 1/2
d²V/dx² = 24x - 24 < 0 for x = 1/2
Hence maximum volume
Volume = ( 3 - 2x)²x
x = 1/2 => 2x =1
= ( 3 - 1)²(1/2)
= 2 m³
maximum volume of the tank = 2 m³
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