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The problem shown discusses bacterial growth modeled by the function:

\[
N(t) = 300 \left(\frac{4}{3}\right)^t
\]

Where:
- \(N(t)\) is the number of bacteria at time \(t\) minutes.
- \(300\) is the initial number of bacteria.
- \(\left(\frac{4}{3}\right)^t\) represents the exponential growth factor.

### 1. **¿Cuántas bacterias están presentes al inicio?**
To find the number of bacteria at the beginning (\(t = 0\)):

\[
N(0) = 300 \left(\frac{4}{3}\right)^0 = 300 \times 1 = 300
\]

Thus, at \(t = 0\) (initial time), there are **300 bacteria**.

### 2. **En forma aproximada, ¿Cuántas bacterias están presentes después de 3 minutos?**
To find the number of bacteria after \(t = 3\) minutes:

\[
N(3) = 300 \left(\frac{4}{3}\right)^3 = 300 \times \left(\frac{64}{27}\right)
\]

Now calculate \(N(3)\):

\[
N(3) = 300 \times 2.3704 \approx 711.11
\]

So, after 3 minutes, there are approximately **711 bacteria**.

---

Would you like further explanation or have any other questions?

### Related Questions:
1. What is the growth rate per minute?
2. How many bacteria will there be after 5 minutes?
3. What is the time it takes for the bacteria population to double?
4. How does the growth factor \(\frac{4}{3}\) affect the bacterial increase rate?
5. How would the function change if the initial number of bacteria was different?

### Tip:
Exponential growth occurs when the growth rate is proportional to the current amount. This means the more bacteria there are, the faster the population grows!

Ejemplo: El número de bacterias presentes en un cultivo después de t minutos está dado por N(t) = 300 (4/3)^t. Observe que N(t) es un múltiplo constante de la función exponencial (4/3)^t. 1. ¿Cuántas bacterias están presentes al inicio? 2. En forma aproximada, ¿Cuántas bacterias están presentes después de 3 minutos?

Explore a bacterial growth problem modeled by the exponential function N(t) = 300 (4/3)^t. Learn how to calculate the initial number of bacteria and estimate the population after 3 minutes. This problem is suitable for students in Grades 9-12 studying exponential growth and functions.

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