Lynbrook West, an apartment complex, has 100 two-bedroom units. The monthly profit (in dollars) realized from renting out x apartments is given by the following function.
P(x) = -10 x^2 + 1780 x - 54,000
To maximize the monthly rental profit, how many units should be rented out?
units
What is the maximum monthly profit realizable?
$
To maximize the monthly rental profit, we need to find the maximum point of the profit function P(x) = -10x^2 + 1780x - 54,000.
To find the maximum point, we can use calculus. The maximum occurs at the vertex of the parabola, which is given by x = -b / (2a), where P(x) is in the form ax^2 + bx + c.
In this case, a = -10, b = 1780, and c = -54,000. Plugging these values into the formula, we have x = -1780 / (2*(-10)) = 89.
Since we are dealing with apartment units, we can only rent out whole units, so rounding up, we should rent out 90 units.
To find the maximum monthly profit realizable, we substitute x = 90 into the profit function P(x):
P(90) = -10 * (90^2) + 1780 * 90 - 54000
= -10 * 8100 + 160200 - 54000
= -81000 + 106200 - 54000
= 22200
Therefore, the maximum monthly profit realizable is $22,200.