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An exponential function is a type of mathematical function that models data where the growth rate is proportional to the current value. In simpler terms, it means that as something grows, its growth becomes faster over time. The general form of an exponential growth function is:
\( P(t) = P_0 \times r^t \)
Here's what each part means:

• \( P(t) \): Population at time \( t \)
• \( P_0 \): Initial population
• \( r \): Growth rate
• \( t \): Time

For our tuberculosis culture problem, we know the initial concentration \( P_0 \) is 5000 cells per ml and the growth rate \( r \) is 1.185. The exponential function becomes:

\( P(t) = 5000 \times 1.185^t \)
This helps us understand how the population of cells increases over time.

The growth rate is a critical part of an exponential function. It's the factor by which the quantity multiplies over a fixed period of time. In this context, the growth rate tells us how quickly the population of the tuberculosis culture is increasing.

In our problem, the growth rate \( r \) is 1.185. This indicates that each hour, the number of tuberculosis cells multiplies by 1.185. To see how this works in practice, consider:

• After 1 hour, the concentration is \( 5000 \times 1.185^1 \)
• After 2 hours, the concentration is \( 5000 \times 1.185^2 \)

As you can see, the concentration gets bigger and bigger, faster and faster. This rapid increase is a hallmark of exponential growth.

Population concentration refers to the number of individuals (or cells) within a specific volume at a given time. For instance, in our tuberculosis culture example, it is initially \( 5000 \) cells per ml. As time progresses, this concentration changes based on the exponential growth rate.

After determining the exponential function: \( P(t) = 5000 \times 1.185^t \), we can calculate the population concentration at any given time \( t \). For example, after 8 hours:

\( P(8) = 5000 \times 1.185^8 \)
Using a calculator, this becomes:

\( P(8) \approx 5000 \times 4.304 \approx 21520 \) cells per ml.

This means that after 8 hours, the concentration of the tuberculosis culture has increased to 21520 cells per ml!