IDNLearn.com is designed to help you find reliable answers quickly and easily. Our platform offers reliable and detailed answers, ensuring you have the information you need.
Sagot :
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.
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.
Thank you for using this platform to share and learn. Don't hesitate to keep asking and answering. We value every contribution you make. Discover the answers you need at IDNLearn.com. Thank you for visiting, and we hope to see you again for more solutions.