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The problem states that a library charges a fine starting after 21 days, with an initial fine of $0.25 for the first day, which doubles every day. We need to determine how many days it will take for the total fine to reach $4.00.

Let's break it down:

1. **First day**: The fine is $0.25.
2. **Second day**: The fine doubles to $0.50.
3. **Third day**: The fine doubles again to $1.00.
4. After this point, the fines continue to double each day.

This doubling pattern suggests an exponential growth. The total fine after $n$ days can be calculated as:

\[
\text{Total fine} = 0.25 + 0.50 + 1.00 + \dots
\]

This sequence follows the powers of 2, where the fine for the $n$-th day is $ 0.25 \times 2^{(n-1)} $.

We need to find the maximum number of days such that the total fine does not exceed $4.00.

I'll calculate the total accumulated fine day by day until it reaches or exceeds $4.00.After calculating the fines day by day:

1. **Day 1**: $0.25
2. **Day 2**: $0.50
3. **Day 3**: $1.00
4. **Day 4**: $2.00
5. **Day 5**: $4.00

The total fine accumulates to $4.00 at the end of **Day 5**. Therefore, the maximum number of days a book may be borrowed before accumulating a total fine of $4.00 is **25 days** (since fines begin after day 21, and this happens on the 5th day of fines).

Thus, the correct answer is **D) 25 days**.

Would you like more details on this, or do you have any questions?

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Here are 5 related questions for further understanding:
1. How would the total fine change if the starting fine was $0.50 instead of $0.25?
2. What if the fine doubled every 2 days instead of daily? How would that affect the total fine over time?
3. Can you find the general formula for the total fine after $n$ days if the fine keeps doubling?
4. How would the answer change if the total fine allowed was increased to $8.00?
5. What is the role of exponential growth in real-world situations, such as compound interest or population growth?

**Tip**: Exponential growth leads to rapid increases over time, which is why doubling patterns can lead to large numbers quickly.

A library charges a fine if a book is not returned after 21 days. The fine is $0.25 the first day, and doubles every following day such that the total fine would be $1.00 after the third day. What is the maximum number of days that a book may be borrowed from the library before accumulating a total fine of $4.00?

Understand how to solve an exponential fine growth problem, where the fine doubles each day after 21 days. Learn how to calculate the total fine accumulation until it reaches $4.00. Suitable for students in Grades 7-9.

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