. An open tank with a square base and vertical sides is to be constructed from a metalsheet so as to hold a given quantity of water. Show that the total surface area is least when depth of the tank is half its width.(2010c)
Let the length, width and height of the open tank be x, x and y units respectively.
Volume = x 2 y
Total surface area = x 2 + 4xy.
Volume of the tank is given to be constant
Now, surface area = x 2 + 4xy
For total surface area to be least,
Hence, surface area is minimum when x = 2y, i.e., the depth of the tank is half of its width.