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The problem involves an exponential growth model for bacteria, with the formula given as:

\[
P = C(2)^{rt}
\]

Where:
- \(P\) is the number of bacteria after \(t\) hours,
- \(C\) is the initial number of bacteria,
- \(r\) is the growth rate,
- \(t\) is time in hours.

### Information given:
- Initially, there are \(12,000\) bacteria, so \(C = 12,000\).
- After \(4\) hours, the number of bacteria has doubled to \(24,000\).
- We need to find the growth rate \(r\).

### Step-by-Step Solution:

1. **Set up the equation with given values**:
   After 4 hours (\(t = 4\)), the number of bacteria doubles, so \(P = 24,000\). Plugging into the formula:
   
   \[
   24,000 = 12,000 \cdot (2)^{4r}
   \]

2. **Simplify the equation**:
   \[
   \frac{24,000}{12,000} = (2)^{4r}
   \]
   \[
   2 = (2)^{4r}
   \]

3. **Solve for \(r\)**:
   \[
   2 = 2^{4r}
   \]
   By comparing the exponents, we get:
   \[
   4r = 1
   \]
   \[
   r = \frac{1}{4}
   \]

Thus, the value of \(r\) is **\(\frac{1}{4}\)**, which corresponds to option **B**.

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Would you like a detailed explanation or have any further questions?

Here are some related questions:
1. How does exponential growth differ from linear growth?
2. What happens to the population if \(r\) is negative in exponential growth?
3. How would the equation change if the bacteria tripled instead of doubled?
4. What would be the formula if the population grew continuously (with base \(e\)) instead of base 2?
5. What is the doubling time formula in terms of the growth rate \(r\)?

**Tip**: Always check if you can simplify the equations by canceling out terms to make solving for unknowns easier!

A scientist initially measures 12,000 bacteria in a growth medium. 4 hours later, the scientist measures 24,000 bacteria. Assuming exponential growth, the formula P = C(2)^(rt) gives the number of bacteria in the growth medium, where r and C are constants and P is the number of bacteria t hours after the initial measurement. What is the value of r?

Learn how to calculate the growth rate of bacteria using exponential growth. A detailed step-by-step solution shows how the formula P = C(2)^(rt) is used to find the growth rate (r), given that the number of bacteria doubles from 12,000 to 24,000 in 4 hours.

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