Question
The amount of money in a savings account over time is given in the table below. Answer the questions below to determine what kind of function would best fit the data, linear or exponential.
Number of Years Since Account Opened, x 11 22 33 44
Money in account ($), f(x) 230, point, 3, 0230.30 491, point, 6, 0491.60 750, point, 8, 2750.82 1, comma, 009, point, 5, 21,009.52
(a linear or an exponential) function would better model the data because as x increases, the y values change (additively or multiplicatively) The (common difference/slope or common ratio/multiplier/base) of this function is approximately (???)
Number of Years Since Account Opened, x 11 22 33 44
Money in account ($), f(x) 230, point, 3, 0230.30 491, point, 6, 0491.60 750, point, 8, 2750.82 1, comma, 009, point, 5, 21,009.52
(a linear or an exponential) function would better model the data because as x increases, the y values change (additively or multiplicatively) The (common difference/slope or common ratio/multiplier/base) of this function is approximately (???)
Answers
GPT-4o mini
To determine whether a linear or exponential function fits the given data, we need to analyze how the values change as the number of years increases. Here’s the data provided:
| Years (x) | Money in account (f(x)) |
|------------|-------------------------|
| 11 | 230.30 |
| 22 | 491.60 |
| 33 | 750.82 |
| 44 | 1,009.52 |
### Step 1: Check for a linear relationship (common difference)
To check for linearity, we calculate the differences in \( f(x) \):
1. \( f(22) - f(11) = 491.60 - 230.30 = 261.30 \)
2. \( f(33) - f(22) = 750.82 - 491.60 = 259.22 \)
3. \( f(44) - f(33) = 1,009.52 - 750.82 = 258.70 \)
The differences are not constant, which means the data is **not linear**.
### Step 2: Check for an exponential relationship (common ratio)
To check for an exponential relationship, we can calculate the ratios of consecutive amounts:
1. \( \frac{f(22)}{f(11)} = \frac{491.60}{230.30} \approx 2.14 \)
2. \( \frac{f(33)}{f(22)} = \frac{750.82}{491.60} \approx 1.53 \)
3. \( \frac{f(44)}{f(33)} = \frac{1,009.52}{750.82} \approx 1.34 \)
These ratios are not constant either, but we can look for a common growth pattern.
### Step 3: Approximate growth factor (approaching exponential)
Instead of strict ratios, we can assess approximate growth:
- \( 491.60 \approx 2.14 \cdot 230.30 \)
- \( 750.82 \approx 1.53 \cdot 491.60 \)
- \( 1,009.52 \approx 1.34 \cdot 750.82 \)
Since the ratios are not exact, we can summarize that the account's growth is accelerating, which corresponds to an exponential relationship.
### Conclusion
Therefore, a(n) **exponential** function would better model the data because as \( x \) increases, the \( f(x) \) values change (multiplicatively).
The approximate growth rate (i.e., base) is not constant but can be roughly averaged based on the ratios we've calculated. If we were to determine a rough average for modeling purposes, we could summarize that the base is around **1.5**, acknowledging it’s not perfectly consistent.
So, summarizing your request:
- An **exponential** function would best model the data.
- The **common ratio/base** of this function is approximately **1.5** (as an average estimate).
| Years (x) | Money in account (f(x)) |
|------------|-------------------------|
| 11 | 230.30 |
| 22 | 491.60 |
| 33 | 750.82 |
| 44 | 1,009.52 |
### Step 1: Check for a linear relationship (common difference)
To check for linearity, we calculate the differences in \( f(x) \):
1. \( f(22) - f(11) = 491.60 - 230.30 = 261.30 \)
2. \( f(33) - f(22) = 750.82 - 491.60 = 259.22 \)
3. \( f(44) - f(33) = 1,009.52 - 750.82 = 258.70 \)
The differences are not constant, which means the data is **not linear**.
### Step 2: Check for an exponential relationship (common ratio)
To check for an exponential relationship, we can calculate the ratios of consecutive amounts:
1. \( \frac{f(22)}{f(11)} = \frac{491.60}{230.30} \approx 2.14 \)
2. \( \frac{f(33)}{f(22)} = \frac{750.82}{491.60} \approx 1.53 \)
3. \( \frac{f(44)}{f(33)} = \frac{1,009.52}{750.82} \approx 1.34 \)
These ratios are not constant either, but we can look for a common growth pattern.
### Step 3: Approximate growth factor (approaching exponential)
Instead of strict ratios, we can assess approximate growth:
- \( 491.60 \approx 2.14 \cdot 230.30 \)
- \( 750.82 \approx 1.53 \cdot 491.60 \)
- \( 1,009.52 \approx 1.34 \cdot 750.82 \)
Since the ratios are not exact, we can summarize that the account's growth is accelerating, which corresponds to an exponential relationship.
### Conclusion
Therefore, a(n) **exponential** function would better model the data because as \( x \) increases, the \( f(x) \) values change (multiplicatively).
The approximate growth rate (i.e., base) is not constant but can be roughly averaged based on the ratios we've calculated. If we were to determine a rough average for modeling purposes, we could summarize that the base is around **1.5**, acknowledging it’s not perfectly consistent.
So, summarizing your request:
- An **exponential** function would best model the data.
- The **common ratio/base** of this function is approximately **1.5** (as an average estimate).