An open tank with a square base and vertical sides is to be constructed from a metal sheet so as to hold a given quantity of water. Show that the cost of material will be least when the depth of the tank is half of its width. If the cost is to be borne by nearby settled lower-income families, for whom water will be provided, what kind of value is hidden in this question?

#### Solution

The base of the tank is square.

Let the length, width and height of the open tank be x, x and y units respectively.

Volume = Length × Breadth × Height = x^{2} y

Total surface area = 2(lb + bh + hl) − lb = x^{2} + 4xy.

The volume of the tank is given to be constant

Now, surface area = x^{2} + 4xy

For the total surface area to be least

Hence, the surface area is minimum when x = 2y, i.e., the depth of the tank is half of its width.

Now if the surface area of the sheet is minimum the cost of the sheet will be least as well, Thus making the tank economical and cost-effective.