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

So the volume V of the tank is given be V = x

^{2}y

And V= 500

So 500 = x

^{2}y

y = 500/x

^{2}(1)

And area of metal sheet required for making this tank is S = x

^{2}+ 4xy ( It is base and four side walls)

So S = x

^{2}+ 4x( 500/x

^{2}) = x

^{2}+ 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 = x

^{2}+ 2000x

^{-1}= 100 + 200 = 300m

^{2}

And y = 500/x

^{2}= 5 m

Hence the height of the tank should be 5m.

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