A bacteria culture is known to grow at a rate proportional to the amount present. After one hour, 1000 strands of the bacteria are observed in the culture; and after four hours, 3000 strands. Find the approximate number of strands of the bacteria originally in the culture.

The growth of the bacteria culture can be modeled using an exponential growth equation. The general form of the equation is:

\[ N(t) = N_0 e^{kt} \]

where:
- \( N(t) \) is the number of bacteria at time \( t \),
- \( N_0 \) is the initial number of bacteria,
- \( k \) is the growth constant,
- \( t \) is the time.

We are given two pieces of information:
1. After 1 hour (\( t = 1 \)), \( N(1) = 1000 \).
2. After 4 hours (\( t = 4 \)), \( N(4) = 3000 \).

First, we use the information for \( t = 1 \):

\[ 1000 = N_0 e^k \]

Next, we use the information for \( t = 4 \):

\[ 3000 = N_0 e^{4k} \]

We can solve these equations simultaneously to find \( N_0 \) and \( k \). First, divide the second equation by the first equation:

\[ \frac{3000}{1000} = \frac{N_0 e^{4k}}{N_0 e^k} \]

This simplifies to:

\[ 3 = e^{3k} \]

Taking the natural logarithm of both sides:

\[ \ln(3) = 3k \]

Solving for \( k \):

\[ k = \frac{\ln(3)}{3} \]

Now we know \( k \), we can substitute it back into the first equation to solve for \( N_0 \):

\[ 1000 = N_0 e^{\frac{\ln(3)}{3}} \]

Simplify the exponent:

\[ e^{\frac{\ln(3)}{3}} = 3^{\frac{1}{3}} \]

Thus, we have:

\[ 1000 = N_0 \cdot 3^{\frac{1}{3}} \]

\[ N_0 = \frac{1000}{3^{\frac{1}{3}}} \]

\[ N_0 \approx \frac{1000}{1.4422} \]

\[ N_0 \approx 693.4 \]

So, the approximate number of strands of the bacteria originally in the culture is around 693.

Would you like further details or have any questions?

Here are 8 related questions for further exploration:
1. What is the general solution to the exponential growth equation?
2. How does the value of the growth constant \( k \) affect the growth rate?
3. How would the initial number of bacteria change if the observed bacteria after one hour were different?
4. What is the significance of the natural logarithm in solving exponential growth problems?
5. How can we verify the accuracy of our solution?
6. How would the growth model change if the rate of growth was not proportional to the amount present?
7. What are some real-world applications of exponential growth models?
8. How can we apply this method to determine the initial amount in other growth contexts, like finance or population studies?

**Tip:** When working with exponential growth or decay problems, always check the consistency of your results by substituting them back into the original equations.

Learn how to solve exponential growth problems using the example of bacteria culture growth. Calculate the approximate number of bacteria originally in the culture based on observed growth rates over time. Suitable for advanced high school students and beyond.

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