We are given that the initial amount of bacteria is $q$ and the amount of bacteria triples every 15 seconds. We are asked to find an expression for the amount of bacteria after $n$ minutes.

Step 1: Conversion of Time

Since the bacteria triples every 15 seconds, we need to express $n$ minutes in terms of seconds. There are 60 seconds in a minute, so: $n \text{ minutes} = 60n \text{ seconds}$

Step 2: Number of 15-second intervals

The bacteria triples every 15 seconds, so the number of 15-second intervals in $60n$ seconds is: $\frac{60n}{15} = 4n$ Thus, the bacteria triples $4n$ times over the course of $n$ minutes.

Step 3: Exponential Growth

Each time the bacteria triples, it is multiplied by 3. After $4n$ intervals, the bacteria will have been tripled $4n$ times. Therefore, the total amount of bacteria after $n$ minutes is: $q \times 3^{4n}$ Thus, the expression for the amount of bacteria after $n$ minutes is: $\boxed{q \times 3^{4n}}$

Would you like more details or have any questions about this? Here are five related questions to deepen your understanding:

1. How would the formula change if the bacteria doubled instead of tripled every 15 seconds?
2. What happens to the bacteria count if $n = 0$?
3. How does the amount of bacteria change if the tripling happens every 30 seconds instead of 15 seconds?
4. Can you express the formula if the bacteria grows by a factor of 4 every 20 seconds?
5. What would be the expression if the initial amount of bacteria were 5 times greater?

Tip: Always pay attention to the time units in growth problems; converting between seconds, minutes, or hours can change the structure of your formula.