Question

A bacteria has an initial population of 500 and is doubling every hour. The bacteria reaches a population of 16,000 bacteria.

Which equation could represent the situation described and how long does it take for the bacteria to reach 16,000?

Answer 1

To model the growth of the bacteria, we can use the equation for exponential growth, which is given by:

\[ P(t) = P_0 \times 2^{t/T} \]

Where:
- \( P(t) \) is the population at time \( t \),
- \( P_0 \) is the initial population,
- \( T \) is the doubling time (in hours),
- \( t \) is the time in hours.

In this scenario:
- The initial population \( P_0 = 500 \).
- The doubling time \( T = 1 \) hour.

Thus the equation becomes:

\[ P(t) = 500 \times 2^t \]

We want to find out how long it takes for the bacteria to reach a population of 16,000. We set \( P(t) = 16,000 \) and solve for \( t \):

\[
500 \times 2^t = 16,000
\]

To solve for \( t \), we first divide both sides by 500:

\[
2^t = \frac{16,000}{500}
\]

Calculating the right side:

\[
\frac{16,000}{500} = 32
\]

Now we have:

\[
2^t = 32
\]

Since \( 32 = 2^5 \), we can equate the exponents:

\[
t = 5
\]

Therefore, it will take **5 hours** for the bacteria to reach a population of 16,000.

In summary, the equation representing the situation is:

\[ P(t) = 500 \times 2^t \]

And it takes **5 hours** to reach a population of 16,000.