In the​ following, write an expression in terms of the given variables that represents the indicated quantity. The amount of bacteria after n min if the initial amount of bacteria is q and the amount of bacteria triples every 15 sec.​ (Hint: The answer should contain q as well as​ n) The expression for the amount of bacteria is

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.

In the following, write an expression in terms of the given variables that represents the indicated quantity. The amount of bacteria after n min if the initial amount of bacteria is q and the amount of bacteria triples every 15 sec. (Hint: The answer should contain q as well as n) The expression for the amount of bacteria is

Learn how to express the growth of bacteria over time using exponential functions. This problem explores how the amount of bacteria, initially q, triples every 15 seconds, and provides the formula for the quantity of bacteria after n minutes: q × 3^(4n). Suitable for high school students studying exponential growth and algebra.

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