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The given population function for the microorganism is:

\[
P(t) = 3.86 \times 10^5 \times 2^{t/4}
\]

We are tasked with finding how long it will take for the population to **quadruple**. 

Let the initial population be \( P_0 = 3.86 \times 10^5 \). Quadrupling means that the population becomes \( 4P_0 \).

Thus, we want to solve for \( t \) when \( P(t) = 4P_0 \).

The equation becomes:

\[
4 \times (3.86 \times 10^5) = 3.86 \times 10^5 \times 2^{t/4}
\]

Simplifying:

\[
4 = 2^{t/4}
\]

We can express 4 as a power of 2:

\[
4 = 2^2
\]

Thus, we have:

\[
2^2 = 2^{t/4}
\]

Since the bases are the same, we can equate the exponents:

\[
2 = \frac{t}{4}
\]

Multiplying both sides by 4:

\[
t = 8
\]

Therefore, it will take **8 hours** for the population to quadruple.

Would you like any further details or explanations? Here are five related questions you could explore:

1. What if the population were to double instead of quadruple? How long would that take?
2. How would the time change if the growth rate exponent \( t/4 \) was different?
3. What is the general form for the time required for the population to increase by a factor \( k \)?
4. How does the population change over time graphically?
5. What are the assumptions behind exponential growth models, and are they always valid?

**Tip:** Exponential growth models are common in biology, but real-life factors often limit unlimited growth. Always consider environmental constraints.

The number of population of a certain microorganism after t hours is 3.86(10^5)(2^(t/4)). How long will it take for the population to quadruple?

Learn how to determine the time it takes for a population of microorganisms to quadruple using an exponential growth function P(t) = 3.86 × 10^5 × 2^(t/4). This solution involves logarithmic equations and exponential growth analysis, suitable for students in Grades 9-12.

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