Question

The manager of a large apartment complex knows from experience that 120 units will be occupied if the rent is 396 dollars per month. A market survey suggests that, on the average, one additional unit will remain vacant for each 3 dollar increase in rent. Similarly, one additional unit will be occupied for each 3 dollar decrease in rent. What rent should the manager charge to maximize revenue?

Answer 1

Answer on Question #37385 – Math - Calculus

Let $n$ be a number of $3 in rent (up or down) and$R$ be a manager's revenue.

For increase in rent:

$$R = (120 - n)(396 + 3n)$$ $$R = -3n^2 - 36n + 47520$$ $$\frac{R}{3} = -n^2 - 12n + 15840$$

and, decrease in rent:

$$R = (120 + n)(396 - 3n)$$ $$R = -3n^2 + 36n + 47520$$ $$\frac{R}{3} = -n^2 + 12n + 15840$$

For rent increase,

$$\left(\frac{R}{3}\right)' = -2n - 12 = 0$$ $$2n = -12$$ $$n_{max} = -6$$ $$\frac{R}{3} = -(-6)^2 - 12 \cdot (-6) + 15840$$ $$\frac{R}{3} = 15876$$

For rent decrease,

$$\left(\frac{R}{3}\right)' = -2n + 12 = 0$$ $$2n = 12$$ $$n_{max} = 6$$ $$\frac{R}{3} = -6^2 + 12 \cdot 6 + 15840$$ $$\frac{R}{3} = 15876$$

Answer: the manager should either increase the rent by $6 \cdot 3 = 18$ dollars, or he should decrease the rent by 18 dollars, either a rent of $396 + 18 = 414$ dollars or $396 - 18 = 378$ dollars will maximize revenue.