We can break down the problem step by step.

Part (a): Exponential Model

The general exponential growth model is given by the equation:

$n(t) = n_0 \cdot 2^{t/a}$

where:

• $n(t)$ is the number of bacteria at time $t$,
• $n_0$ is the initial number of bacteria,
• $t$ is the time (in hours),
• $a$ is the doubling time (in hours).

From the problem, we know:

• The initial number of bacteria, $n_0 = 25$,
• The doubling time, $a = 5$ hours.

Thus, the model becomes:

$n(t) = 25 \cdot 2^{t/5}$

Part (b): Estimate the Number of Bacteria After 18 Hours

Now, we use the model from part (a) to estimate the number of bacteria after $t = 18$ hours. Substituting $t = 18$ into the equation:

$n(18) = 25 \cdot 2^{18/5}$

First, calculate $18/5$:

$18/5 = 3.6$

Now, calculate $2^{3.6}$ (using a calculator):

$2^{3.6} \approx 12.32$

Finally, multiply by 25:

$n(18) = 25 \cdot 12.32 \approx 308$

So, the estimated number of bacteria after 18 hours is approximately 308.

Part (c): Find When the Bacteria Count Reaches 2 Million

We need to find the time $t$ when the bacteria count $n(t) = 2,000,000$. Set up the equation:

$2,000,000 = 25 \cdot 2^{t/5}$

First, divide both sides by 25:

$80,000 = 2^{t/5}$

Now, take the logarithm of both sides (base 2):

$\log_2(80,000) = \frac{t}{5}$

We can convert this to base 10 logarithms using the identity $\log_2(x) = \frac{\log_{10}(x)}{\log_{10}(2)}$:

$\log_2(80,000) = \frac{\log_{10}(80,000)}{\log_{10}(2)}$

Using a calculator, we find:

$\log_{10}(80,000) \approx 4.903 \quad \text{and} \quad \log_{10}(2) \approx 0.3010$

Thus:

$\log_2(80,000) = \frac{4.903}{0.3010} \approx 16.28$

Now, substitute this back into the equation:

$16.28 = \frac{t}{5}$

Multiply by 5:

$t = 16.28 \times 5 = 81.4$

So, the bacteria count will reach 2 million in approximately 81.4 hours.

Summary:

• (a) The exponential model is $n(t) = 25 \cdot 2^{t/5}$.
• (b) After 18 hours, the estimated number of bacteria is approximately 308.
• (c) The bacteria count will reach 2 million in approximately 81.4 hours.

Do you want more details on any of the steps or calculations?

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Here are 5 related questions for further practice:

1. If the bacteria doubled every 3 hours instead of 5, how would the model change?
2. How long would it take for the bacteria to reach 1 million?
3. What would the bacteria count be after 10 hours?
4. How would you modify the equation if the initial count of bacteria was 50 instead of 25?
5. What happens to the time required to reach 2 million if the doubling time increases to 6 hours?

Tip:

In exponential growth problems, logarithms are very useful for solving time-related questions when the population reaches a certain size!