To find an exponential regression equation for the given data and determine the number of bacteria after 16 hours, we'll follow these steps:

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Step 1: Exponential Regression Equation

The general form of an exponential equation is:
$y = a \cdot b^x$
where:

• $y$ is the dependent variable (bacteria count),
• $x$ is the independent variable (time in hours),
• $a$ is the initial value (bacteria at $x = 0$),
• $b$ is the growth factor per hour.

We'll use the given data to compute the values of $a$ and $b$.

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Step 2: Use the Data for Exponential Regression

The data:

x &: 0, 1, 2, 3, 4, 5 \\ y &: 8989, 9999, 108108, 122122, 132132, 139139 \\ \end{aligned}$$ Using these values, an exponential regression equation can be calculated. I'll do this step now. numpy as np from scipy.optimize import curve_fit # Given data x = np.array([0, 1, 2, 3, 4, 5]) # Time in hours y = np.array([8989, 9999, 108108, 122122, 132132, 139139]) # Bacteria count # Exponential function definition def exp_func(x, a, b): return a * b**x # Fit the data to the exponential function params, _ = curve_fit(exp_func, x, y) # Extract the coefficients a and b a, b = params # Predict the number of bacteria after 16 hours x_predict = 16 y_predict = exp_func(x_predict, a, b) a_rounded, b_rounded, y_predict_rounded = round(a, 2), round(b, 2), round(y_predict) a_rounded, b_rounded, y_predict_rounded### **Exponential Regression Equation** The exponential regression equation for the given data is: $$y = 39659.98 \cdot 1.32^x$$ --- ### **Step 3: Predict the Number of Bacteria After 16 Hours** Using the regression equation $$y = 39659.98 \cdot 1.32^x$$, substituting $$x = 16$$, we get: $$y = 39659.98 \cdot 1.32^{16} \approx 3,454,301$$ Thus, the number of bacteria present after 16 hours is approximately: $$\boxed{3,454,301}$$ --- ### Would you like a detailed breakdown or clarification? Here are **5 follow-up questions** to expand your understanding: 1. How is the growth factor $$b$$ derived in an exponential regression? 2. What does the initial value $$a$$ represent in this context? 3. How does this model differ from a linear regression for the same data? 4. Can this equation be used for very large $$x$$ values? Why or why not? 5. What would happen if the bacteria grew at a linear rate instead of exponentially? --- **Tip**: When working with exponential growth, always check if the model fits the data well before making long-term predictions.