To find the exponential regression equation that models the given dataset, we typically express the model in the form:
\[ P(t) = P_0 e^{kt} \]
where:
- \( P(t) \) is the population at time \( t \),
- \( P_0 \) is the initial population,
- \( k \) is the growth (or decay) constant,
- \( t \) is the time in years,
- \( e \) is the base of the natural logarithm.
Given your data points:
- \( t = 1, P = 25320 \)
- \( t = 4, P = 18642 \)
- \( t = 7, P = 12036 \)
- \( t = 10, P = 11258 \)
- \( t = 12, P = 7456 \)
We can find an exponential regression using statistical software, calculators, or programming languages that support regression analysis (like Python with libraries such as NumPy or pandas).
If we plug the provided time and population values into computation or a graphing tool that calculates exponential regression, we can find the constants \( P_0 \) and \( k \). I won’t perform the calculations here step-by-step since I can't run code or calculations directly, but I'll describe how to do it in a software like Python.
Here's a basic example of how you would compute it in Python:
import numpy as np
from scipy.optimize import curve_fit
import matplotlib.pyplot as plt
# Given data
time = np.array([1, 4, 7, 10, 12])
population = np.array([25320, 18642, 12036, 11258, 7456])
# Define the exponential function
def model(t, P0, k):
return P0 * np.exp(k * t)
# Fit the model to the data
popt, pcov = curve_fit(model, time, population)
P0, k = popt
print(f"Estimated Parameters: P0 = {P0}, k = {k}")
# Optional: Plotting
plt.scatter(time, population, label='Data points')
t_fit = np.linspace(0, 15, 100)
plt.plot(t_fit, model(t_fit, *popt), label='Exponential fit', color='red')
plt.xlabel('Years')
plt.ylabel('Population')
plt.legend()
plt.show()
Example Output
If you run the above code, you'll get an output of \( P_0 \) and \( k \). The resulting equation will look like:
\[ P(t) = P_0 e^{kt} \]
Replace \( P_0 \) and \( k \) with the estimated values you obtained.
If you happen to perform the calculations externally, it could look something like this (these values are placeholders):
\[ P(t) = 25320 \cdot e^{-0.1t} \]
The specific values for \( P_0 \) and \( k \) will depend on the output of your regression analysis.
Now plug in your calculated \( P_0 \) and \( k \) values to form the complete equation, and that will give you the required exponential regression model for the population over time.