Let's break down the problem using the table in the image. The table represents Amanda's bank account balance over time:

• Time (months): 6, 8, 10, 12
• Money (dollars): 460, 330, 200, 70

Part (a) – Relation between time and money

Looking at the trend in the table, as time increases, the amount of money in her account decreases. Therefore, the correct option would be:

• "As time increases, the amount of money in her account decreases."

We are also asked to determine the rate at which the money is decreasing. The rate of change (slope) between two points $(x_1, y_1)$ and $(x_2, y_2)$ can be found using the formula: $\text{Rate} = \frac{y_2 - y_1}{x_2 - x_1}$ Let's calculate the rate of decrease between any two points. We'll use the points (6, 460) and (8, 330) to calculate the rate: $\text{Rate} = \frac{330 - 460}{8 - 6} = \frac{-130}{2} = -65 \, \text{dollars per month}$

Thus, the money is decreasing at a rate of 65 dollars per month.

Part (b) – Initial amount in the account

To find how much money Amanda had in her account before she started withdrawing money, we need to extend the relationship back to month 0. Given that the account balance decreases linearly, we can use the slope-intercept form of the equation for a line: $y = mx + b$, where $m$ is the rate of change and $b$ is the initial amount.

From part (a), we know that the rate of change $m = -65$. Let's use the point (6, 460) to solve for $b$:

$460 = -65(6) + b$ $460 = -390 + b$ $b = 460 + 390 = 850$

Thus, Amanda had 850 dollars in her bank account before she started withdrawing money.

Let me know if you'd like further details, and here are some related questions to practice:

1. How would the equation of the line look if Amanda deposited money instead of withdrawing?
2. How would the equation change if Amanda withdrew at a different rate?
3. What would the balance be at month 14?
4. How could you determine the total amount Amanda withdrew after 12 months?
5. How can you generalize this approach for different starting amounts and withdrawal rates?

Tip: Linear relationships, like this one, are often represented by straight lines, where the slope indicates how one variable changes with respect to another.