Question

A billboard is to be made with 100m^2 of printed area and with margins of 2m at the top and bottom and 4m on each side. Find the outside dimensions of the billboard if its total area is to be minimum.

Answer 1

Length of billboard = x
Width of billboard = (100/x)

Length including margin = x + 4 + 4 = (x+8)
Width including margin = (100/x) + 2 + 2 = (100/x + 4)

The total area is given by:

(x+8)*(100/x + 4)
= 100 + (800/x) + 4x + 32
= 4x + (800/x) + 132

This is the function to be minimized. The derivative of the function is:

f'(x) = 4 - (800/x^2)

Equating this to zero,

4 = (800/x^2)
x^2 = 200
x = √200
= 10√2

This is the required value of 'x' at the minima (the negative value is irrelevant for this problem)

Put the value of x to get the required dimensions.