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The rate of increase of bacteria in certain culture is proportional to the number of bacteria present. if it is found to be number double in 5 hour, prove that the bacteria became a 8 times at the end of 15 hour.?
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The rate of increase of bacteria in certain culture is proportional to...
For every 5hrs it doubles I.e. up to 15 hrs it means, n=3
2^n=2^3=8....
2^n=2^3=8....
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The rate of increase of bacteria in certain culture is proportional to...
Understanding Bacterial Growth
The growth of bacteria can be modeled by the equation:
- dN/dt = kN
where:
- N is the number of bacteria,
- k is a constant of proportionality,
- t is time.
Doubling Time
It is given that the bacteria double in 5 hours. Mathematically, this can be expressed as:
- N(5) = 2N(0)
This indicates that the bacteria population at time 5 hours is twice the initial population.
Exponential Growth Model
Using the principle of exponential growth, we can express the number of bacteria at any time t:
- N(t) = N(0) * e^(kt)
From the doubling time, we know:
- N(5) = N(0) * e^(5k) = 2N(0)
This implies:
- e^(5k) = 2
Taking the natural logarithm:
- 5k = ln(2)
- k = ln(2) / 5
Calculating Population at 15 Hours
To find the number of bacteria at 15 hours:
- N(15) = N(0) * e^(15k)
Substituting k:
- N(15) = N(0) * e^(15 * (ln(2) / 5))
- N(15) = N(0) * e^(3ln(2))
- N(15) = N(0) * (e^(ln(2)))^3
- N(15) = N(0) * 2^3
- N(15) = N(0) * 8
Conclusion
Thus, at the end of 15 hours, the bacteria population becomes 8 times the initial population, proving the assertion.
The growth of bacteria can be modeled by the equation:
- dN/dt = kN
where:
- N is the number of bacteria,
- k is a constant of proportionality,
- t is time.
Doubling Time
It is given that the bacteria double in 5 hours. Mathematically, this can be expressed as:
- N(5) = 2N(0)
This indicates that the bacteria population at time 5 hours is twice the initial population.
Exponential Growth Model
Using the principle of exponential growth, we can express the number of bacteria at any time t:
- N(t) = N(0) * e^(kt)
From the doubling time, we know:
- N(5) = N(0) * e^(5k) = 2N(0)
This implies:
- e^(5k) = 2
Taking the natural logarithm:
- 5k = ln(2)
- k = ln(2) / 5
Calculating Population at 15 Hours
To find the number of bacteria at 15 hours:
- N(15) = N(0) * e^(15k)
Substituting k:
- N(15) = N(0) * e^(15 * (ln(2) / 5))
- N(15) = N(0) * e^(3ln(2))
- N(15) = N(0) * (e^(ln(2)))^3
- N(15) = N(0) * 2^3
- N(15) = N(0) * 8
Conclusion
Thus, at the end of 15 hours, the bacteria population becomes 8 times the initial population, proving the assertion.
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The rate of increase of bacteria in certain culture is proportional to the number of bacteria present. if it is found to be number double in 5 hour, prove that the bacteria became a 8 times at the end of 15 hour.?
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The rate of increase of bacteria in certain culture is proportional to the number of bacteria present. if it is found to be number double in 5 hour, prove that the bacteria became a 8 times at the end of 15 hour.? for Class 12 2024 is part of Class 12 preparation. The Question and answers have been prepared according to the Class 12 exam syllabus. Information about The rate of increase of bacteria in certain culture is proportional to the number of bacteria present. if it is found to be number double in 5 hour, prove that the bacteria became a 8 times at the end of 15 hour.? covers all topics & solutions for Class 12 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The rate of increase of bacteria in certain culture is proportional to the number of bacteria present. if it is found to be number double in 5 hour, prove that the bacteria became a 8 times at the end of 15 hour.?.
The rate of increase of bacteria in certain culture is proportional to the number of bacteria present. if it is found to be number double in 5 hour, prove that the bacteria became a 8 times at the end of 15 hour.? for Class 12 2024 is part of Class 12 preparation. The Question and answers have been prepared according to the Class 12 exam syllabus. Information about The rate of increase of bacteria in certain culture is proportional to the number of bacteria present. if it is found to be number double in 5 hour, prove that the bacteria became a 8 times at the end of 15 hour.? covers all topics & solutions for Class 12 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The rate of increase of bacteria in certain culture is proportional to the number of bacteria present. if it is found to be number double in 5 hour, prove that the bacteria became a 8 times at the end of 15 hour.?.
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