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The question asks about bacteria growth over time, and here's how to approach it:

### Part (a): Exponential Model
The amount of bacteria triples every 15 days, and there are 500 bacteria at time \( t = 0 \). This indicates an exponential growth pattern. The general formula for exponential growth is:

\[
N(t) = N_0 \cdot r^{\frac{t}{T}}
\]

Where:
- \( N(t) \) is the number of bacteria at time \( t \).
- \( N_0 \) is the initial number of bacteria (500 in this case).
- \( r \) is the growth factor (since it triples, \( r = 3 \)).
- \( T \) is the time it takes to triple (15 days).
- \( t \) is the time in days.

So, the exponential model is:

\[
N(t) = 500 \cdot 3^{\frac{t}{15}}
\]

### Part (b): Number of Bacteria After 40 Days
To find how many bacteria are present after 40 days, substitute \( t = 40 \) into the model:

\[
N(40) = 500 \cdot 3^{\frac{40}{15}} = 500 \cdot 3^{2.67}
\]

Calculating \( 3^{2.67} \), you get approximately:

\[
N(40) \approx 500 \cdot 12.23 = 6115
\]

So, there are approximately **6,115 bacteria** after 40 days.

Would you like more details or have any questions?

### Related Questions:
1. How would the model change if the bacteria doubled instead of tripled?
2. What if the growth period was 10 days instead of 15?
3. Can this exponential model be used for other biological populations? Why or why not?
4. How would you model this scenario if the growth slowed down over time?
5. What are some factors that could limit the exponential growth of bacteria?

### Tip:
Exponential growth often appears in real-world phenomena like populations or investments, but it eventually slows due to limiting factors (resources, space).

At time t = 0, 500 bacteria are in a petri dish, and this amount triples every 15 days. 
(a) Give an exponential model for the situation. 
(b) How many bacteria are there in the dish after 40 days?

Learn how to model exponential growth in a bacteria population that triples every 15 days, starting with 500 bacteria. Find the number of bacteria after 40 days using the exponential growth formula.

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