Question

A square tank of capacity 250 cubic metres has to be dug out. The cost of the land is Rs 50 per square meter. The cost of digging increases with the depth and for the whole tank it is Rs 400*h2 where h meters is the depth of the tank. what should be the dimensions of the tank so that the cost be minimum?

Answer 1

Let thedimension of square tank be 'x' (length) and 'x' (breadth)and depthis given as 'h' . So volume of tank 250 = x2*h which impliesx2 = 250/h ...(i)

cost of land be Z = 50x2

Further, cost of digging is= 400 h2

So total cost of tankf(h)=cost of land + cost of digging =50 x2 +400 h2 OR

f(h)= 50* 250/h +400 h2 .....(putting x2 = 250/h from (i) )

Now putting f'(h) = 0 gives -12500/h2 + 800 h =0 = h3 = 12500/800 = 125/8 = h = 5/2

further f''(h) becomes positive so h = 5/2 is minima.

hnce for getting minimum value of f(h) (i.e. cost of tank) depth h = 5/2 Metre

further x2 = 250/h =100 sqM and x = 10 Metres. So dimension of the tank are 10 (length)x 10(breadth) x 5/2 (height)