The table shows the fines for overdue library books based on the number of days overdue. To find the initial value (the fine at day 0), we can examine the fines for the days listed and see if we can extrapolate backward.
The fines per day can be determined by looking at the increases:
- From day 2 to day 3: \( 45 \text{ cents} - 30 \text{ cents} = 15 \text{ cents} \)
- From day 3 to day 4: \( 60 \text{ cents} - 45 \text{ cents} = 15 \text{ cents} \)
Thus, the fine increases by 15 cents each day. To find the fine for day 0, we can subtract 15 cents from the fine on day 2 (30 cents):
\[ \text{Fine on day 0} = 30 \text{ cents} - 15 \text{ cents} = 15 \text{ cents} \]
So, the initial value (the fine owed for day 0) is 15 cents.