To find the rate of change in terms of balance (in dollars) for the number of weeks, we can calculate the change in balance for a given change in weeks (i.e., the slope of the line connecting the points).
The formula for the rate of change (slope) is given by:
\[ \text{Rate of Change} = \frac{\Delta \text{Balance}}{\Delta \text{Weeks}} = \frac{B_2 - B_1}{W_2 - W_1} \]
where \( B_1 \) and \( B_2 \) are the balances at weeks \( W_1 \) and \( W_2 \), respectively.
Let's calculate the rate of change between some of the points.
-
Between Week 0 and Week 1:
- \( W_1 = 0 \), \( B_1 = 32 \)
- \( W_2 = 1 \), \( B_2 = 39 \)
- \[ \text{Rate of Change} = \frac{39 - 32}{1 - 0} = \frac{7}{1} = 7 \text{ dollars/week} \]
-
Between Week 1 and Week 3:
- \( W_1 = 1 \), \( B_1 = 39 \)
- \( W_2 = 3 \), \( B_2 = 53 \)
- \[ \text{Rate of Change} = \frac{53 - 39}{3 - 1} = \frac{14}{2} = 7 \text{ dollars/week} \]
-
Between Week 3 and Week 6:
- \( W_1 = 3 \), \( B_1 = 53 \)
- \( W_2 = 6 \), \( B_2 = 74 \)
- \[ \text{Rate of Change} = \frac{74 - 53}{6 - 3} = \frac{21}{3} = 7 \text{ dollars/week} \]
-
Between Week 6 and Week 8:
- \( W_1 = 6 \), \( B_1 = 74 \)
- \( W_2 = 8 \), \( B_2 = 88 \)
- \[ \text{Rate of Change} = \frac{88 - 74}{8 - 6} = \frac{14}{2} = 7 \text{ dollars/week} \]
-
Between Week 8 and Week 13:
- \( W_1 = 8 \), \( B_1 = 88 \)
- \( W_2 = 13 \), \( B_2 = 213 \)
- \[ \text{Rate of Change} = \frac{213 - 88}{13 - 8} = \frac{125}{5} = 25 \text{ dollars/week} \]
From these calculations, we see that the rate of change is consistent at $7 per week for the first four intervals (0 to 8 weeks). However, from week 8 to week 13, the rate of change increases significantly to $25 per week.
Thus, the overall rates of change indicate:
- From Week 0 to Week 8: $7 per week
- From Week 8 to Week 13: $25 per week
In summary, the general rate of change up to week 8 is $7 per week, while it increases dramatically after that point.