Question

The cost of building an office complex, x floors high, in a prime location in Accra is made up of three components:

GH¢10 million for the land

GH¢1/4 million per floor

Specialized costs of GH¢10000x per floor.

How many floors should the office complex contain if the average cost per floor is to be minimized?

Answer 1

Average cost per floor $ATC$ is

$$ATC=\dfrac{TC}{x}=\dfrac{10}{x}+0.25+0.01x ,$$

million per floor

Marginal cost is

$$MC=(10+0.25x+0.01x^2)'=0.25+0.02x$$

million per floor

Average cost per floor $ATC$ is minimized if $ATC=MC$

$$\dfrac{10}{x}+0.25+0.01x =0.25+0.02x,\ x>0$$

$$\dfrac{10}{x}-0.01x =0$$

$$1000-x^2=0$$

$$x=\sqrt{1000}$$

$x=31$

$$ATC=\dfrac{10}{31}+0.25+0.01(31)=0.8826$$

$x=32$

$$ATC=\dfrac{10}{32}+0.25+0.01(32)=0.8825$$

The office complex should contain 32 floors if the average cost per floor is to be minimized.