Cost minimization problem

Space for a square tank of capacity 250 cubic metres has to be dug out. The cost of the land is 50 dollars per square metre. The cost of digging increases with the depth and for the whole tank it is 400*h^2 dollars where h metres is the depth of the tank. What should be the dimensions of the tank so that the cost incurred is a minimum?

Student X

Let each side of the square tank be x.

Then volume of tank is

(x^2)(h)= 250 =======> x^2 =250/h ----------(1)

Cost of land intended for tank occupation = 50*x^2

Total cost incurred C= 50*x^2 + 400*h^2

Substituting in (1), we have

C= 50* (250/h)+ 400*h^2 =12500/h + 400*h^2

When cost is a minimum, dC/dh =0 

     ie  -12500/(h^2) + 800h =0 

           12500/(h^2)= 800h 

            h^3 = 15.625 =====> h= 2.5 ,    x=sqrt(250/2.5) =10

The required dimensions of the tank for cost minimization are therefore that of a square base with side 10m and height 2.5m. (shown)

Hope this helps. Peace.

Best Regards,

Mr Koh