A scientist starts with 80 bacteria. After 8 hours, she has 160 bacteria. 
Assuming the bacteria are growing exponentially,

1) Find the growth rate

2) Find an exponential equation that gives the population of bacteria, y, after x hours. Make sure to have at least 3 digits after the decimal.

We are given that the bacteria population grows exponentially, which means the population \( y \) after \( x \) hours can be modeled by the equation:

\[
y = y_0 \cdot e^{kx}
\]
where:
- \( y_0 \) is the initial population,
- \( k \) is the growth rate,
- \( x \) is time in hours, and
- \( y \) is the population after \( x \) hours.

### 1) Finding the Growth Rate
Given:
- Initial population, \( y_0 = 80 \) bacteria,
- Population after 8 hours, \( y(8) = 160 \) bacteria.

Substitute into the exponential growth formula:

\[
160 = 80 \cdot e^{8k}
\]

Solve for \( k \):

\[
\frac{160}{80} = e^{8k} \implies 2 = e^{8k}
\]

Take the natural logarithm of both sides:

\[
\ln(2) = 8k
\]

Thus:

\[
k = \frac{\ln(2)}{8}
\]

Let’s calculate \( k \).

\[
k \approx \frac{0.693}{8} = 0.0866 \, \text{(rounded to 4 decimal places)}.
\]

### 2) Finding the Exponential Equation
Now that we have the growth rate \( k \approx 0.0866 \), the exponential equation for the bacteria population after \( x \) hours is:

\[
y = 80 \cdot e^{0.0866x}
\]

This equation gives the population of bacteria at any time \( x \), with three decimal places for the growth rate.

Would you like more details on how to interpret the equation, or any clarifications?

Here are five related questions:
1. How would the population change after 24 hours using this equation?
2. What is the doubling time of the bacteria population?
3. How can the equation be adjusted if a different growth rate is observed?
4. What does the growth rate \( k \) represent biologically?
5. How would the equation change if the initial population was different?

**Tip:** When dealing with exponential growth, the natural logarithm is often key to isolating variables like growth rate.

A scientist starts with 80 bacteria. After 8 hours, she has 160 bacteria. Assuming the bacteria are growing exponentially, 1) Find the growth rate 2) Find an exponential equation that gives the population of bacteria, y, after x hours. Make sure to have at least 3 digits after the decimal.

Learn how to calculate the growth rate and write the exponential equation for bacterial growth, starting with 80 bacteria and doubling to 160 in 8 hours. This step-by-step solution explores key concepts like exponential growth and logarithms. Suitable for students in Grades 10-12.

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