An open tank is to be constructed with a square based and vertical sides so as to contain 500 cube metres of water. What should be the dimension of the tank, if the area of metal sheet used in its construction is to be minimum

Let the square base of the tank be of side x and the height be h
So the volume V of the tank is given be V  = x2y
And V= 500
So 500 = x2y
y = 500/x2  (1)

And area of metal sheet required for making this tank is S = x2 + 4xy ( It is base and four side walls)
So S = x2 + 4x( 500/x2) = x2 + 2000x-1
So S' = 2x - 2000x-2
So for critical points, we have S' = 0
So 2x - 2000x-2 = 0
Or x = 10, and x = 0 is not possible.
To check whether x = 10 is minima or maxima find S'' =0 at x = 10, if S" >0, then it is minimum.
So S'' = 2 + 4000x-3
At x = 10 , S" = 6 >0
So x = 10, there exist a minimum value.
So Area of the metal sheet = x2 + 2000x-1 = 100 + 200 = 300m2
And y = 500/x2  = 5 m
Hence the height of the tank should be 5m.



 

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