Let’s solve this step by step.

Problem Setup

The growth rate of bacteria is proportional to the number present, which suggests exponential growth. The equation for exponential growth can be written as:

$P(t) = P_0 e^{kt}$

Where:

• $P(t)$: number of bacteria at time $t$,
• $P_0$: initial population,
• $k$: growth constant,
• $t$: time in minutes.

Given:

1. Initially ($t = 0$), $P_0 = 2000$.
2. At $t_1 = 7$, $P(7) = 6000$.
3. At $t_1 + 14 = 14$, $P(14) = 12000$.

We need to:

1. Find $P(t)$ in terms of $t$,
2. Compute $P(28)$,
3. Solve for $t_1$ when $P(t) = 6000$.

Part (a) - Find $P(t)$

Start with the equation:

$P(t) = 2000 e^{kt}.$

Step 1: Use $P(7) = 6000$ to solve for $k$.

At $t = 7$, $P(7) = 6000$:

$6000 = 2000 e^{7k}.$

Divide through by 2000:

$3 = e^{7k}.$

Take the natural logarithm (ln) of both sides:

$\ln(3) = 7k.$

Solve for $k$:

$k = \frac{\ln(3)}{7}.$

Step 2: Write $P(t)$.

Substitute $k$ back into the exponential equation:

$P(t) = 2000 e^{\frac{\ln(3)}{7} t}.$

Part (b) - Find $P(28)$

Substitute $t = 28$ into $P(t)$:

$P(28) = 2000 e^{\frac{\ln(3)}{7} \cdot 28}.$

Simplify the exponent:

$\frac{\ln(3)}{7} \cdot 28 = 4 \ln(3).$

Thus:

$P(28) = 2000 e^{4 \ln(3)}.$

Using the property $e^{\ln(a^b)} = a^b$:

$P(28) = 2000 \cdot 3^4.$

Simplify $3^4 = 81$:

$P(28) = 2000 \cdot 81 = 162000.$

So:

$P(28) = 162000 \text{ bacteria.}$

Part (c) - Find $t_1$ when $P(t) = 6000$

Use $P(t) = 2000 e^{\frac{\ln(3)}{7} t}$ and substitute $P(t) = 6000$:

$6000 = 2000 e^{\frac{\ln(3)}{7} t}.$

Divide by 2000:

$3 = e^{\frac{\ln(3)}{7} t}.$

Take the natural logarithm of both sides:

$\ln(3) = \frac{\ln(3)}{7} t.$

Solve for $t$:

$t = 7.$

Thus, $t_1 = 7$ minutes.

Final Answers:

1. $P(t) = 2000 e^{\frac{\ln(3)}{7} t}$,
2. $P(28) = 162000$ bacteria,
3. $t_1 = 7$ minutes.

Let me know if you'd like further clarification or details!

Questions:

1. How does exponential growth differ from linear growth?
2. What are the properties of natural logarithms ($\ln(x)$) that make them useful in solving such equations?
3. How does changing the initial population ($P_0$) affect $P(t)$?
4. Can you derive the doubling time for this population using $k$?
5. How does this problem illustrate the proportionality constant's role in population dynamics?

Tip:

Always double-check unit consistency (e.g., time in minutes) to ensure correctness in exponential growth problems.