The number of bacterial colonies doubles every 15 hours. If the number of the bacterial colonies in a petri dish is initially 90, what is the number of bacterial colonies on the 120th hour?

This problem involves exponential growth, where the number of bacterial colonies doubles every 15 hours. The general formula for exponential growth is given by:

\[
N(t) = N_0 \times 2^{\frac{t}{T}}
\]

Where:
- \( N(t) \) is the number of bacterial colonies at time \( t \).
- \( N_0 \) is the initial number of bacterial colonies.
- \( T \) is the doubling time.
- \( t \) is the elapsed time.

For this problem:
- \( N_0 = 90 \)
- \( T = 15 \) hours
- \( t = 120 \) hours

Plugging in these values into the formula:

\[
N(120) = 90 \times 2^{\frac{120}{15}}
\]

Simplifying the exponent:

\[
\frac{120}{15} = 8
\]

So:

\[
N(120) = 90 \times 2^8
\]

Calculating \( 2^8 \):

\[
2^8 = 256
\]

Therefore:

\[
N(120) = 90 \times 256 = 23040
\]

The number of bacterial colonies on the 120th hour is 23,040.

Would you like more details or have any questions? Here are some related questions you might find interesting:

1. What would be the number of colonies after 45 hours?
2. How would the formula change if the doubling time were 20 hours?
3. What if the initial number of colonies were 150 instead of 90?
4. How many hours would it take for the colonies to reach 1,000,000?
5. What is the general form of the exponential growth formula for different growth rates?

**Tip:** When dealing with exponential growth, remember that doubling time directly affects the speed of increase. The smaller the doubling time, the faster the growth rate.

Learn how to calculate the exponential growth of bacterial colonies in a petri dish, starting from 90 colonies, and doubling every 15 hours. Using the formula N(t) = N_0 * 2^(t/T), find the number of colonies after 120 hours. This problem is ideal for students in Grades 9-12.

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