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The problem involves bacterial growth, which doubles every half hour, starting with 300 bacteria.

### (a) How many bacteria are there after 3 hours?

- Since the bacteria double every half hour, the number of bacteria after *t* hours can be modeled by an exponential function. 
- In 3 hours, there are \( 2 \times 3 = 6 \) half-hour periods.
  
The formula to calculate the number of bacteria after 3 hours is:

\[
y = 300 \times 2^{6}
\]

### (b) How many bacteria are there after *t* hours?

- The general form for bacterial growth in this case is:

\[
y = 300 \times 2^{2t}
\]

- Here, \( t \) is the number of hours, and the factor \( 2t \) accounts for the bacteria doubling every half hour.

### (c) How many bacteria are there after 40 minutes?

- 40 minutes is \( \frac{2}{3} \) of an hour, so the number of half-hour periods is \( 2 \times \frac{2}{3} = \frac{4}{3} \).
  
Thus, the formula for 40 minutes becomes:

\[
y = 300 \times 2^{\frac{4}{3}}
\]

Now let's calculate these values.

I'll compute them.Here are the results for each part of the problem:

### (a) After 3 hours:
- The number of bacteria is \( 19,200 \).

### (b) General formula:
- For any number of hours \( t \), the number of bacteria is given by \( y = 300 \times 2^{2t} \).
  - After 3 hours, this also results in \( 19,200 \) bacteria, confirming part (a).

### (c) After 40 minutes:
- The number of bacteria is approximately \( 756 \) (rounded to the nearest whole number).

Would you like further details or clarification on any part of the solution?

Here are 5 related questions for practice:
1. How many bacteria are there after 1 hour?
2. How many half-hour periods are there in 5 hours?
3. If the bacteria double every 15 minutes, how would the equation change?
4. What is the population after 6 hours?
5. How long will it take for the population to reach 100,000?

**Tip:** For exponential growth problems, focus on understanding the doubling time and how it influences the time intervals in your formula!

A bacterial culture starts with 300 bacteria and doubles in size every half hour. How many bacteria are there after 3 hours? What is the general formula to calculate the bacteria after t hours? How many bacteria are there after 40 minutes?

This problem involves calculating bacterial growth over time. Starting with 300 bacteria, the culture doubles every half hour. Learn how to find the number of bacteria after 3 hours, how to derive the general formula for t hours, and calculate the population after 40 minutes. Suitable for students in Grades 9-12.

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