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The given problem describes the rate of change of a bacterial population over time using the function \( P'(t) = 3\sqrt{t} - \frac{9}{\sqrt{t}} + 2 \), where \( P(t) \) represents the population at time \( t \).

### Known Data:
- The population at \( t = 9 \) is given as 14.
- We are asked to find the population at \( t = 16 \).

### Approach:
1. **Determine \( P(t) \) (population function):** We need to integrate the given rate of change function \( P'(t) \) to find the general form of \( P(t) \).

   \( P'(t) = 3\sqrt{t} - \frac{9}{\sqrt{t}} + 2 \)

   - For \( 3\sqrt{t} \), the integral is \( 3 \times \frac{2}{3} t^{3/2} = 2t^{3/2} \).
   - For \( -\frac{9}{\sqrt{t}} \), the integral is \( -9 \times 2t^{1/2} = -18t^{1/2} \).
   - For the constant \( 2 \), the integral is \( 2t \).

   Thus, the population function is:
   \[
   P(t) = 2t^{3/2} - 18t^{1/2} + 2t + C
   \]
   where \( C \) is the constant of integration.

2. **Use initial condition to find \( C \):**
   We know that \( P(9) = 14 \), so substitute \( t = 9 \) into the equation:
   \[
   14 = 2(9)^{3/2} - 18(9)^{1/2} + 2(9) + C
   \]
   - \( (9)^{3/2} = 27 \)
   - \( (9)^{1/2} = 3 \)

   Substituting these values:
   \[
   14 = 2(27) - 18(3) + 18 + C
   \]
   \[
   14 = 54 - 54 + 18 + C
   \]
   \[
   14 = 18 + C \quad \Rightarrow \quad C = -4
   \]

   So, the population function is:
   \[
   P(t) = 2t^{3/2} - 18t^{1/2} + 2t - 4
   \]

3. **Find \( P(16) \):**
   Now, substitute \( t = 16 \) into the equation to find the population at \( t = 16 \):
   \[
   P(16) = 2(16)^{3/2} - 18(16)^{1/2} + 2(16) - 4
   \]
   - \( (16)^{3/2} = 64 \)
   - \( (16)^{1/2} = 4 \)

   Substituting these values:
   \[
   P(16) = 2(64) - 18(4) + 32 - 4
   \]
   \[
   P(16) = 128 - 72 + 32 - 4
   \]
   \[
   P(16) = 84
   \]

Thus, the population at time \( t = 16 \) is **84**, which corresponds to option **C**.

Would you like a deeper explanation or any further details?

Here are 5 related questions to explore:

1. How do you integrate functions involving square roots?
2. What other methods could we use to find constants in differential equations?
3. Can population growth always be modeled by a differential equation like this one?
4. What happens if the initial population condition is unknown? How can we handle such cases?
5. How can we approximate bacterial population growth using discrete models?

**Tip:** When solving for a constant of integration, always ensure that you're applying any initial or boundary conditions accurately, as they are key to determining the specific solution.

A doctor is studying the growth of a bacterial colony over time. The rate of change of the population at any time t (in hours) is modeled by the function: P'(t) = 3√t − 9/√t + 2. The population of the bacteria at time t = 9 is 14. Find the population of the bacteria at time t = 16.

Learn how to solve a bacterial population growth problem modeled by a differential equation. Using integration, we calculate the population at t = 16, with known population data at t = 9. Suitable for high school and early college students.

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