Question: Suppose that a library contains $12,000$ books now, and adds $250$ books every month.
(a) Write an explicit formula for a linear model that describes this growth.
$P_n =$
(b) If this trend continues, how many books will the library contain in a year?
books
(c) How long will it be before the library contains $19,000$ books?
Round to the nearest month.
months
1. **State the problem:**
We have a library with an initial number of books $12,000$ and it adds $250$ books every month. We want to model this growth linearly.
2. **Write the explicit formula:**
The general formula for linear growth is:
$$P_n = P_0 + dn$$
where:
- $P_n$ is the number of books after $n$ months,
- $P_0$ is the initial number of books,
- $d$ is the monthly increase,
- $n$ is the number of months.
Given $P_0 = 12,000$ and $d = 250$, the formula is:
$$P_n = 12,000 + 250n$$
3. **Calculate the number of books after 1 year (12 months):**
Substitute $n = 12$:
$$P_{12} = 12,000 + 250 \times 12$$
$$P_{12} = 12,000 + 3,000 = 15,000$$
So, after 1 year, the library will have $15,000$ books.
4. **Find how long until the library has $19,000$ books:**
Set $P_n = 19,000$ and solve for $n$:
$$19,000 = 12,000 + 250n$$
Subtract $12,000$ from both sides:
$$19,000 - 12,000 = 250n$$
$$7,000 = 250n$$
Divide both sides by $250$:
$$n = \frac{7,000}{250}$$
Show cancellation:
$$n = \frac{\cancel{7,000}}{\cancel{250}} = 28$$
So, it will take $28$ months for the library to have $19,000$ books.
**Final answers:**
(a) $$P_n = 12,000 + 250n$$
(b) After 1 year, the library will have $15,000$ books.
(c) It will take $28$ months to reach $19,000$ books.