Question

A real estate company is going to build a new residential complex. The land they have purchased can hold at most 500 apartments. Also, if they make x apartments, then the monthly maintenance cost for the whole complex would be as follows:

Fixed cost = ₹ 4000

Variable cost = ₹ (14x – 0.04x²)

How many apartments should the complex have in order to minimize the maintenance costs?

Answer 1

Given, C(x) = 4000 + 14x – 0.04x², 0 ≤ x ≤ 500

Since the coefficient of x² is negative (−0.04), the curve opens downward, so the minimum cost will occur at the end points of the interval.

C(0) = 4000 + 14 × 0 – 0.04 × (0)² = 4000

C(500) = 4000 + 14 × 500 – 0.04 × (500)²

= 4000 + 7000 – 10000

= 11000 – 10000

= 1000

C(500) < C(0)

Minimum maintenance cost occurs when x = 500 apartments.