To solve the problem of determining how long it will take for a population of bacteria to grow from 500 to 4000 given that it doubles every 3 hours, we can use the concept of exponential growth and logarithms. Here's the step-by-step solution:
1. Understanding the Exponential Growth Formula:
The population growth can be modeled using the formula:
[tex]\[
P(t) = P_0 \times 2^{(t/T)}
\][/tex]
where:
- [tex]\( P(t) \)[/tex] is the population at time [tex]\( t \)[/tex],
- [tex]\( P_0 \)[/tex] is the initial population,
- [tex]\( T \)[/tex] is the doubling time,
- [tex]\( t \)[/tex] is the time elapsed.
2. Substitute the Values:
Given:
- Initial population, [tex]\( P_0 = 500 \)[/tex],
- Target population, [tex]\( P(t) = 4000 \)[/tex],
- Doubling time, [tex]\( T = 3 \)[/tex] hours.
The equation becomes:
[tex]\[
4000 = 500 \times 2^{(t/3)}
\][/tex]
3. Isolate the Exponential Term:
Divide both sides of the equation by 500:
[tex]\[
\frac{4000}{500} = 2^{(t/3)}
\][/tex]
Simplify the left side:
[tex]\[
8 = 2^{(t/3)}
\][/tex]
4. Solve for [tex]\( t \)[/tex] Using Logarithms:
To isolate [tex]\( t \)[/tex], take the logarithm base 2 of both sides:
[tex]\[
\log_2(8) = \left(\frac{t}{3}\right)
\][/tex]
We know that [tex]\( \log_2(8) = 3 \)[/tex]:
[tex]\[
3 = \frac{t}{3}
\][/tex]
5. Solve for [tex]\( t \)[/tex]:
Multiply both sides of the equation by 3 to solve for [tex]\( t \)[/tex]:
[tex]\[
t = 3 \times 3 = 9
\][/tex]
Therefore, the time needed for the population to reach 4000 starting from an initial population of 500 is 9 hours.
