Question

a real estate office handles 50 apartment units. when the rent is $580 per month, all units are occupied. for each $40 increase in rent, however, an average of one unit becomes vacant. each occupied unit requires an average of $45 per month for service and repairs. find the monthly profit function and determine its domain.

Answer 1

The monthly profit function for the real estate office is P(x) = (50 - x)(580 + 40x) - 45(50 - x), with the domain being 0 ≤ x ≤ 50, where x represents the number of $40 rent increases.

Step-by-step explanation:

To find the monthly profit function for a real estate office with a rent price and vacancy rate relationship, let's define x as the number of $40 rent increases from the initial $580. The total number of rented units will be 50 - x, as for each $40 increase, one unit becomes vacant.

The monthly rent income is calculated by the product of the number of units rented and the rent price per unit, which increases by $40x from the initial amount. So, the monthly rent income is (50 - x)(580 + 40x).

The cost associated with service and repairs each month for the occupied units is $45 per unit, which gives us a monthly cost of 45(50 - x).

Putting it all together, the monthly profit function, P(x), is the total income minus the total costs, which is: P(x) = (50 - x)(580 + 40x) - 45(50 - x). The domain of this function is the set of values of x for which the function makes sense, i.e., when the number of units rented is non-negative, which is 0 ≤ x ≤ 50.