To determine the rate of change of the account balance over time, we need to look at how much the balance changes for each day that passes. This is done by calculating the slope of the linear function that models the account balances over time.
Let's summarize the information from the table:
[tex]\[
\begin{array}{|c|c|}
\hline Time \, (x) \, \text{in} \, \text{days} & Account \, Balance \, (y) \, \text{in} \, \text{dollars} \\
\hline 1 & 15 \\
\hline 3 & 7 \\
\hline 5 & -1 \\
\hline 10 & -21 \\
\hline
\end{array}
\][/tex]
To find the rate of change, we use the slope formula between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex]:
[tex]\[
\text{slope} = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
By selecting the initial point [tex]\((x_1, y_1) = (1, 15)\)[/tex] and the final point [tex]\((x_2, y_2) = (10, -21)\)[/tex], we can substitute these values into the slope formula:
[tex]\[
\text{slope} = \frac{-21 - 15}{10 - 1} = \frac{-36}{9} = -4.0
\][/tex]
Interpreting this rate of change, which is [tex]\(-4.0\)[/tex], it means that for each day that passes, the account balance decreases by 4 dollars.
Therefore, the correct interpretation is:
[tex]\[
-4, \text{ the account loses four dollars each day}
\][/tex]
