To answer these questions, let's examine the relationship between the time and the amount of money in Susan's bank account.
(a) First, we need to determine how the amount of money in the account changes with time.
Looking at the table, we have:
- At 6 months, the money in the account is [tex]$476.
- At 8 months, the money in the account is $[/tex]606.
- At 10 months, the money in the account is [tex]$736.
- At 12 months, the money in the account is $[/tex]866.
From these data points, it's clear that as time increases, the amount of money in the account also increases. The appropriate statement is:
"As time increases, the amount of money in the account increases."
Next, we need to determine the rate at which the amount of money increases. This can be done by calculating the change in money over the change in time.
[tex]\[
\text{Rate of increase per month} = \frac{\text{Change in money}}{\text{Change in time}}
\][/tex]
Using the data from the table, we calculate:
[tex]\[
\text{Change in money} = \$866 - \$476 = \$390
\][/tex]
[tex]\[
\text{Change in time} = 12 \text{ months} - 6 \text{ months} = 6 \text{ months}
\][/tex]
[tex]\[
\text{Rate of increase per month} = \frac{\$390}{6 \text{ months}} = \$65 \text{ dollars per month}
\][/tex]
Thus, the rate at which the amount of money in the account is increasing is [tex]$65$[/tex] dollars per month.
(b) To find how much money was already in the account when Susan started depositing money, we need to determine the initial amount of money in the account at time zero (0 months).
Using the linear relationship [tex]\( y = mx + b \)[/tex], where:
- [tex]\( y \)[/tex] is the money in the account,
- [tex]\( m \)[/tex] is the rate of increase ([tex]$65$[/tex] dollars per month),
- [tex]\( x \)[/tex] is the time in months,
- [tex]\( b \)[/tex] is the initial amount of money in the account.
From the table, using any data point, say at 6 months:
[tex]\[
476 = 65 \cdot 6 + b
\][/tex]
Solving for [tex]\( b \)[/tex]:
[tex]\[
476 = 390 + b
\][/tex]
[tex]\[
b = 476 - 390
\][/tex]
[tex]\[
b = 86
\][/tex]
Therefore, the initial amount of money in the account when Susan started depositing money was [tex]$86$[/tex] dollars.
