Question

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A storage tank is to be made from a sheet of metal, 9 meters square, by cutting a square of side x meters from each corner and bending up the sides. What is the maximum volume of the tank, if the height of the tank must not exceed 3 meters ?​

Answer 1

Answer:

9×9 = 81

3×3 = 9

Now - Total - Perimeter

81-9 = 72

Answer 2

Answer:

Maximum volume of the tank =  2  m³

Step-by-step explanation:

Assuming open Tank :

Metal sheet of metal 9 meters square

So the side of  sheet metal = 3 m

A square cut form each side  = x m

Then Base =  (3  - 2x ) m

Height = 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 - 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 =  ( 3 - 2x)²x

=  ( 3 - 2×1/2)²(1/2)

=  2  m³

Maximum volume of the tank =  2  m³

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