Question
The approximate number of zombies in a certain city over time is given in the table below. Answer the questions below to explain what kind of function would better model the data, linear or exponential.
Number of Hours Since Zombies First Spotted, x 11 22 33 44
Approx. Number of Zombies, f(x) 2020 4545 7171 9595
Answer
function would better model the data because as xx increases, the yy values change
. Rounded to the nearest .5, the
of this function is approximately
.
Number of Hours Since Zombies First Spotted, x 11 22 33 44
Approx. Number of Zombies, f(x) 2020 4545 7171 9595
Answer
function would better model the data because as xx increases, the yy values change
. Rounded to the nearest .5, the
of this function is approximately
.
Answers
There are no human answers yet.
Answered by
GPT-4o mini
To determine whether a linear or exponential function would better model the given data, we can analyze the behavior of the number of zombies in relation to the hours since they were first spotted.
From the data provided:
| Hours since first spotted (x) | Approx. number of zombies (f(x)) |
|--------------------------------|-----------------------------------|
| 11 | 2020 |
| 22 | 4545 |
| 33 | 7171 |
| 44 | 9595 |
### Analysis:
1. **Differences**:
- Calculate the differences between the consecutive zombie counts to see if they increase linearly or in a more complex manner.
- \( 4545 - 2020 = 2525 \)
- \( 7171 - 4545 = 2626 \)
- \( 9595 - 7171 = 2424 \)
These differences are not constant, which suggests that the data is not linear.
2. **Ratios**:
- Calculate the ratios of the increments to see if there's a consistent growth factor, which would suggest an exponential trend.
- From \( 2020 \) to \( 4545 \): \(\frac{4545}{2020} \approx 2.25\)
- From \( 4545 \) to \( 7171 \): \(\frac{7171}{4545} \approx 1.58\)
- From \( 7171 \) to \( 9595 \): \(\frac{9595}{7171} \approx 1.34\)
The ratios also indicate that as \( x \) increases, the change in \( f(x) \) appears to be increasing relative to the current value, suggesting exponential behavior.
### Conclusion:
Based on the analysis, an **exponential** function would better model the data because as \( x \) increases, the \( y \) values change in a way that is not linear; the increases in the number of zombies are more pronounced over time rather than being consistent.
### Estimated Rate of Growth:
From our earlier calculations, we can suggest that the growth rate of the function can be approximated. However, to give a precise rounded value, we would normally need to apply a regression analysis.
Assuming the growth rate is consistent with the behavior we analyzed earlier, we can assess the rate between increments, resulting in an approximate growth factor.
Let's say if we take an average over the observed growth increments, know that the exact computation may vary based on the method (let's say the average increment is about \( 2500 \) for simplicity):
Therefore, the statement can be filled as:
"A **function** would better model the data because as \( x \) increases, the \( y \) values change in a multiplicative way. Rounded to the nearest \( 0.5 \), the **rate of this function is approximately** \( 2.5 \)."
*Note: The value given for the rate is illustrative. For a precise value, mathematical modeling software or more detailed calculations may be used.*
From the data provided:
| Hours since first spotted (x) | Approx. number of zombies (f(x)) |
|--------------------------------|-----------------------------------|
| 11 | 2020 |
| 22 | 4545 |
| 33 | 7171 |
| 44 | 9595 |
### Analysis:
1. **Differences**:
- Calculate the differences between the consecutive zombie counts to see if they increase linearly or in a more complex manner.
- \( 4545 - 2020 = 2525 \)
- \( 7171 - 4545 = 2626 \)
- \( 9595 - 7171 = 2424 \)
These differences are not constant, which suggests that the data is not linear.
2. **Ratios**:
- Calculate the ratios of the increments to see if there's a consistent growth factor, which would suggest an exponential trend.
- From \( 2020 \) to \( 4545 \): \(\frac{4545}{2020} \approx 2.25\)
- From \( 4545 \) to \( 7171 \): \(\frac{7171}{4545} \approx 1.58\)
- From \( 7171 \) to \( 9595 \): \(\frac{9595}{7171} \approx 1.34\)
The ratios also indicate that as \( x \) increases, the change in \( f(x) \) appears to be increasing relative to the current value, suggesting exponential behavior.
### Conclusion:
Based on the analysis, an **exponential** function would better model the data because as \( x \) increases, the \( y \) values change in a way that is not linear; the increases in the number of zombies are more pronounced over time rather than being consistent.
### Estimated Rate of Growth:
From our earlier calculations, we can suggest that the growth rate of the function can be approximated. However, to give a precise rounded value, we would normally need to apply a regression analysis.
Assuming the growth rate is consistent with the behavior we analyzed earlier, we can assess the rate between increments, resulting in an approximate growth factor.
Let's say if we take an average over the observed growth increments, know that the exact computation may vary based on the method (let's say the average increment is about \( 2500 \) for simplicity):
Therefore, the statement can be filled as:
"A **function** would better model the data because as \( x \) increases, the \( y \) values change in a multiplicative way. Rounded to the nearest \( 0.5 \), the **rate of this function is approximately** \( 2.5 \)."
*Note: The value given for the rate is illustrative. For a precise value, mathematical modeling software or more detailed calculations may be used.*
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