A landlady currently rents each of her 50 apartments for $1300 per month. She estimates that for each $100 increase in rent, two additional apartments will remain vacant.
a. Construct a function that represents the revenue R(n) as a function of the number of rent increases, n. (Hint: Find the rent per unit after n increases and the number of units rented after n increases.)
R(n)= 100n+1300n+2
b. After how many rent increases will all the apartments be empty?
I thought I could equal the function to zero, but I got a weird answer of -0.00142
c. What is a reasonable domain for this function?
I graphed and got -5≤ n ≤ 25
I think my function is wrong since my answers were incorrect.
Please Help.
The function should be:
R(n) = (50-2n)*(1300+100*n)
To construct a function that represents the revenue R(n) as a function of the number of rent increases, n, we need to consider the information given in the problem.
Let's break down the information step-by-step:
1. The current rent for each apartment is $1300 per month.
2. The landlady estimates that for each $100 increase in rent, two additional apartments will remain vacant. This means that for every $100 increase in rent, the number of vacant apartments will increase by 2.
3. We need to find the revenue R(n), which is the total amount of money collected from renting the apartments after n rent increases.
Now, let's determine the rent per unit after n increases:
The initial rent is $1300 per month. For each increase of $100, the rent will increase by $100 x n (since we have n increases). Therefore, the rent per unit after n increases will be: $1300 + ($100 x n).
Next, let's find the number of units rented after n increases:
Given that for each $100 increase in rent, two additional apartments will remain vacant, we can calculate the number of apartments rented as follows:
Total number of apartments = 50 (constant)
Number of apartments rented = (Total number of apartments) - (Number of vacant apartments)
= 50 - (2 x n)
= 50 - 2n
Finally, we can calculate the revenue R(n) as:
R(n) = (Rent per unit after n increases) x (Number of apartments rented)
= ($1300 + ($100 x n)) x (50 - 2n)
= (1300n + 100n^2) - (100n + 800n)
= 100n^2 + 800n - 800n
= 100n^2
Now, let's answer the specific questions:
a. Construct a function that represents the revenue R(n) as a function of the number of rent increases, n.
The correct function is R(n) = 100n^2.
b. After how many rent increases will all the apartments be empty?
To determine when all the apartments will be empty, we need to find when the Number of apartments rented equals 0.
50 - 2n = 0
2n = 50
n = 25
Therefore, after 25 rent increases, all the apartments will be empty.
c. What is a reasonable domain for this function?
The domain represents the set of values that n can take in the context of this problem. Since n represents the number of rent increases, it cannot be negative, as we cannot have a negative number of increases. Additionally, considering the given information, it would be reasonable to assume that there is a limit to the number of increases.
Based on your graph, the domain could be -5 ≤ n ≤ 25. However, since the number of rent increases cannot be negative, the reasonable domain would be n ≥ 0.
In summary:
a. The correct function that represents the revenue R(n) is R(n) = 100n^2.
b. After 25 rent increases, all the apartments will be empty.
c. A reasonable domain for this function would be n ≥ 0.