https://file.aiquickdraw.com/mathbot/file/1727919395067yxjpwsow.jpg

This problem involves exponential growth, as the bacteria are doubling over a fixed period of time.

The formula for exponential growth is:

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

Where:
- \( N(t) \) is the population at time \( t \).
- \( N_0 \) is the initial population.
- \( T \) is the doubling time.
- \( t \) is the elapsed time.

Given:
- Initial population \( N_0 = 100 \).
- Doubling time \( T = 25 \) hours.
- Final population \( N(t) = 190 \).

We need to solve for \( t \), the time it takes to reach 190 bacteria.

1. Set up the equation:
\[
190 = 100 \cdot 2^{\frac{t}{25}}
\]

2. Divide both sides by 100:
\[
1.9 = 2^{\frac{t}{25}}
\]

3. Take the logarithm of both sides (use base 2):
\[
\log_2(1.9) = \frac{t}{25}
\]

4. Solve for \( t \):
\[
t = 25 \cdot \log_2(1.9)
\]

Using the change of base formula for logarithms (\( \log_b(x) = \frac{\log(x)}{\log(b)} \)):

\[
t = 25 \cdot \frac{\log(1.9)}{\log(2)}
\]

Now, let me calculate that for you.The time it would take for the bacterial population to reach 190, to the nearest tenth of an hour, is approximately **23.1 hours**.

Would you like more details on this solution or have any additional questions?

Here are five related questions you could explore:
1. How would the time change if the initial population was 200 instead of 100?
2. What if the bacteria doubled every 20 hours? How would that affect the time to reach 190 bacteria?
3. If we wanted to reach 500 bacteria, how long would that take under the same conditions?
4. How would you model a scenario where the growth rate isn't constant but increases over time?
5. Can you solve a similar problem where bacteria die at a certain rate while still doubling?

**Tip:** Logarithms are very useful for solving exponential growth problems, especially when trying to determine the time in such cases!

In a lab experiment, 100 bacteria are placed in a petri dish. The conditions are such that the number of bacteria is able to double every 25 hours. How long would it be, to the nearest tenth of an hour, until there are 190 bacteria present?

Solve an exponential growth problem involving bacteria in a petri dish. This example covers how to calculate the time it takes for a population of bacteria to grow from 100 to 190, using the exponential growth formula and logarithms. Suitable for high school students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation