Question

The table shows the profit from a school book fair based on the number of books sold.

Books Sold (x) Profit f(x)
10 $50
25 $200
30 $300
35 $350
The rate of change for the function represented by the table is $
per book.

Answer 1

To find the rate of change of the profit with respect to the number of books sold, we can look for the change in profit divided by the change in the number of books sold over the intervals given in the table.

Let's calculate the rate of change between the different intervals:

1. From 10 to 25 books:
- Profit change: \(200 - 50 = 150\)
- Books sold change: \(25 - 10 = 15\)
- Rate of change: \(\frac{150}{15} = 10\) dollars per book.

2. From 25 to 30 books:
- Profit change: \(300 - 200 = 100\)
- Books sold change: \(30 - 25 = 5\)
- Rate of change: \(\frac{100}{5} = 20\) dollars per book.

3. From 30 to 35 books:
- Profit change: \(350 - 300 = 50\)
- Books sold change: \(35 - 30 = 5\)
- Rate of change: \(\frac{50}{5} = 10\) dollars per book.

So, the rate of change varies across the intervals.

To summarize:
- From 10 to 25 books, the rate of change is $10 per book.
- From 25 to 30 books, the rate is $20 per book.
- From 30 to 35 books, the rate is $10 per book.

If you seek an overall representation for the entire dataset, you could take an average across the overall changes.

However, typically for a rate of change, you would consider using the slope which can be calculated as follows if you consider total books sold (10 to 35) and total profit:

- Total profit change from 10 to 35 books:
\[350 - 50 = 300\]

- Total books change from 10 to 35:
\[35 - 10 = 25\]

Thus, the overall average rate of change per book sold is:
\[
\frac{300}{25} = 12 \text{ dollars per book.}
\]

The rate of change represented by the entire data set could be considered as **$12 per book** on average. However, please clarify if you'd like to focus only on specific intervals or the overall average.