Certain bacteria are being cultured in a
laboratory. The number of bacteria t
days after the start of the experiment is
represented by k ∙ 3^t/3 , where k is a
positive constant. Suppose that there are
27,000 bacteria 9 days after the start of
the experiment.
How many bacteria are there at the start
of the experiment?

The number of bacteria \( t \) days after the start of the experiment is given by the function:

\[
N(t) = k \cdot 3^{t/3}
\]

We are given that 9 days after the start of the experiment, the number of bacteria is 27,000. So, we can set up the equation:

\[
N(9) = k \cdot 3^{9/3} = 27,000
\]

Simplifying the exponent:

\[
N(9) = k \cdot 3^3 = k \cdot 27
\]

Now, we can solve for \( k \):

\[
k \cdot 27 = 27,000
\]

\[
k = \frac{27,000}{27} = 1,000
\]

Now that we know \( k = 1,000 \), we can find the number of bacteria at the start of the experiment (\( t = 0 \)):

\[
N(0) = 1,000 \cdot 3^{0/3} = 1,000 \cdot 3^0 = 1,000 \cdot 1 = 1,000
\]

So, there were **1,000 bacteria** at the start of the experiment.

Would you like more details or have any questions?

Here are 5 related questions:
1. How would the number of bacteria change if \( k \) were doubled?
2. What if the growth factor was \( 2 \) instead of \( 3 \)?
3. How many bacteria would be present after 15 days?
4. What does the constant \( k \) represent in this context?
5. How does the exponential growth rate compare to linear growth?

**Tip:** Understanding the base of the exponential function is crucial, as it determines how quickly the population grows over time.

Learn how to calculate the initial number of bacteria in a culture using the exponential growth formula. Given that there are 27,000 bacteria 9 days after the start of the experiment, find the initial count. Understand the exponential growth concept and its application in real-world scenarios. Suitable for Grades 10-12.

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