Let's break down the problem and solve it step by step.

Part (a):

You are asked to show that the population after 3 hours is 3456. The bacteria population grows at a rate of 2% per hour. The initial population is 2000 at time $t = 0$.

We can model exponential growth with the formula: $N(t) = N_0 \cdot (1 + r)^t$ where:

• $N(t)$ is the population at time $t$,
• $N_0$ is the initial population (2000 bacteria),
• $r$ is the growth rate (2% per hour, so $r = 0.02$),
• $t$ is the time in hours.

Substituting the values: $N(3) = 2000 \cdot (1 + 0.02)^3 = 2000 \cdot (1.02)^3$

Now, calculating $(1.02)^3$: $(1.02)^3 \approx 1.061208$ Thus: $N(3) = 2000 \cdot 1.061208 = 3456.42$ Rounding to the nearest whole number, the population after 3 hours is 3456, which matches the expected result. This completes part (a).

Part (b):

The population after $h$ hours is modeled by the formula: $N = a \cdot b^h$ We need to determine the values of $a$ and $b$.

From the previous part, we know:

• $N_0 = 2000$ at $h = 0$,
• $N(3) = 3456$.

Substituting into the formula $N = a \cdot b^h$ when $h = 0$: $2000 = a \cdot b^0 = a \cdot 1$ Thus, $a = 2000$.

To find $b$, use the information that after 3 hours, $N(3) = 3456$: $3456 = 2000 \cdot b^3$ Solving for $b^3$: $b^3 = \frac{3456}{2000} = 1.728$ Taking the cube root of both sides: $b = \sqrt[3]{1.728} \approx 1.2$

So the values are: $a = 2000, \quad b = 1.2$

Part (c):

You are asked to find the value of $k$, where the population after 12 hours is $k$ times the population at time $t = 0$.

The population after 12 hours is: $N(12) = 2000 \cdot (1.2)^{12}$

Calculating $(1.2)^{12}$: $(1.2)^{12} \approx 8.9161$ Thus: $N(12) = 2000 \cdot 8.9161 = 17832.2$

The population at time $t = 0$ was 2000. So $k$ is the ratio of the population at 12 hours to the initial population: $k = \frac{17832.2}{2000} \approx 8.916$

Therefore, $k \approx 8.92$.

Final Answers:

• (a) The population after 3 hours is 3456.
• (b) $a = 2000$, $b = 1.2$.
• (c) $k \approx 8.92$.

Would you like further clarification on any part?

Here are 5 related questions:

1. What does the exponential growth formula imply about population growth over time?
2. How would the population change if the growth rate was 3% instead of 2%?
3. What would happen if the initial population was different, say 3000 bacteria?
4. How does changing the time affect the value of $b$ in the exponential formula?
5. Could this model be applied to other biological growths, and how would we adjust it?

Tip: Always remember to check for rounding in calculations when working with exponential growth, as small differences can change the final results.