A cuboidal shaped godown with square base is to be constructed. Three times as much cost per square meter is incurred for constructing the roof as compared to the walls. Find the dimensions of the godown if it is to enclose a given volume and minimize the cost of constructing the roof and the walls.

As base is square so length of base = breadth of base

Let the length and breadth of the base =x

Also, let the height of the godown = y

Let “C” be the total cost of constructing the godown and “V” be the given volume.

As, it is given that, cost is proportional to the area and three times as much cost per square meter is incurred for constructing the roof as compared to the walls.

Area of roof = x²,

area of 4 walls = 4xy

Therefore,

C = k [3x² + 4xy], … (1)

where k>0 is constant of proportionality

x²y = V … (2)

Substituting value of y from equation (3), in equation (1), we get

On differentiating,

For maximum or minimum value,

So,

When,

Now,

Put, in ,

So,

As, k > 0 so,

Now, as ,

So, C is minimum at

From (3),

So, C is minimum whenand

So,