https://r2.erweima.ai/mathbot/image/17264454784404f9wwn7l.jpg

The problem you're dealing with is related to determining when the concentration of bacteria in a river, represented by the function \( B(t) \), will reduce to a certain level using the Newton-Raphson method.

The equation given is:

\[
B(t) = 75e^{-1.5t} + 20e^{-0.075t}
\]

Where:
- \( B(t) \) represents the concentration of bacteria in thousands of bacteria per liter (mbl).
- \( t \) is the time in hours.

We are tasked to find the time \( t \) when \( B(t) = 12 \) mbl.

### Steps to solve using the Newton-Raphson method:

1. **Rewrite the equation**: First, set the equation equal to the desired concentration (12 mbl):
   \[
   75e^{-1.5t} + 20e^{-0.075t} - 12 = 0
   \]
   Let this equation be \( f(t) \):
   \[
   f(t) = 75e^{-1.5t} + 20e^{-0.075t} - 12
   \]

2. **Compute the derivative \( f'(t) \)**: We need the derivative of \( f(t) \) with respect to \( t \) for the Newton-Raphson method.

3. **Apply the Newton-Raphson formula**:
   \[
   t_{n+1} = t_n - \frac{f(t_n)}{f'(t_n)}
   \]
   Start with an initial guess, likely \( t = 0 \) as suggested.

I will now calculate the derivative \( f'(t) \) and set up the iterative process to find \( t \). Let me compute this.The derivative of the function \( f(t) = 75e^{-1.5t} + 20e^{-0.075t} - 12 \) is:

\[
f'(t) = -112.5e^{-1.5t} - 1.5e^{-0.075t}
\]

Now that we have both \( f(t) \) and \( f'(t) \), we can use the Newton-Raphson method to iterate and find the value of \( t \) where the concentration \( B(t) = 12 \).

I'll perform the iterations starting from \( t = 0 \).It seems like I can’t do more advanced data analysis right now. Please try again later. 

However, you can still follow these steps on your own:

1. Use the Newton-Raphson method with the function \( f(t) = 75e^{-1.5t} + 20e^{-0.075t} - 12 \) and its derivative \( f'(t) = -112.5e^{-1.5t} - 1.5e^{-0.075t} \).
2. Start with an initial guess \( t = 0 \).
3. Apply the formula iteratively:
   \[
   t_{n+1} = t_n - \frac{f(t_n)}{f'(t_n)}
   \]
4. Continue until the difference between successive values of \( t \) is very small (within your tolerance).

Would you like more guidance on performing these calculations manually or using a calculator?

Here are some follow-up questions to expand the understanding:

1. What is the significance of the exponential decay in the bacteria concentration?
2. How does the Newton-Raphson method work for finding roots in general?
3. Could other numerical methods be used to find the value of \( t \)?
4. What would happen to the concentration if the chemical treatment was applied continuously?
5. How can sensitivity analysis be performed on the initial concentration values?

**Tip**: When using the Newton-Raphson method, make sure the derivative does not get too close to zero, as it can lead to inaccuracies or non-convergence.

Determine the time necessary for the concentration of bacteria to reduce to 12 mbl using the Newton-Raphson method, where the concentration is modeled by the equation B = 75e^{-1.5t} + 20e^{-0.075t}.

Learn how to apply the Newton-Raphson method to solve the exponential decay equation B(t) = 75e^{-1.5t} + 20e^{-0.075t}, and determine the time when the bacteria concentration reaches 12 mbl. This guide covers key steps, including derivative computation and iterative calculation, suitable for advanced students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation