Question 1171310: Use a graphing utility to create a scatter diagram of the data given in the table. Observe the shape of the scatter diagram to determine whether the data is best described by an exponential, logarithmic, or logistic model. Then use the appropriate regression feature to find an equation that models the data. When necessary, round values to five decimal places.
x 1 2 3 4 5 6 7 8 9 10
f(x) 409.6 260.4 170.3 110.9 74.1 44.4 32.5 19.2 12.8 8.4
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Absolutely! Let's analyze the data and find the appropriate model.
**1. Create the Scatter Diagram:**
The scatter diagram will help us visualize the relationship between x and f(x).
```python
import matplotlib.pyplot as plt
import numpy as np
x = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]
f_x = [409.6, 260.4, 170.3, 110.9, 74.1, 44.4, 32.5, 19.2, 12.8, 8.4]
plt.scatter(x, f_x)
plt.xlabel('x')
plt.ylabel('f(x)')
plt.title('Scatter Diagram')
plt.grid(True)
plt.show()
```
**2. Observe the Shape:**
The scatter plot shows a decreasing trend, where the rate of decrease slows down as x increases. This suggests an **exponential decay** model.
**3. Find the Exponential Model:**
We will use the exponential regression feature to find the equation that best fits the data.
An exponential model has the form: f(x) = a * e^(-bx)
```python
from scipy.optimize import curve_fit
def exponential_func(x, a, b):
return a * np.exp(-b * x)
popt, pcov = curve_fit(exponential_func, x, f_x)
a_fit = popt[0]
b_fit = popt[1]
print(f"Exponential Model: f(x) = {a_fit:.5f} * e^(-{b_fit:.5f} * x)")
```
**Output:**
Exponential Model: f(x) = 630.48226 * e^(-0.43561 * x)
**4. Conclusion:**
The data is best described by an exponential decay model. The equation that models the data is:
f(x) = 630.48226 * e^(-0.43561 * x)
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