Math Problem Statement
The doubling period of a bacterial population is 15 minutes. At time t = 100 minutes, the bacterial population was 60000. Find the size of the bacterial population after 5 hours.
Solution
The problem you're dealing with involves exponential growth, where the bacterial population doubles every 15 minutes. To solve it, we'll use the exponential growth model formula:
Where:
- is the population at time .
- is the initial population (at ).
- is the doubling time (in this case, 15 minutes).
- is the time elapsed.
Step 1: Find the initial population
At minutes, the bacterial population was 60,000. We can use this information to find .
Substitute into the equation:
Now, simplify the exponent:
So, the equation becomes:
Now, solve for :
Using a calculator for the exponent:
Thus:
So the initial population is approximately 6,608.
Step 2: Find the population after 5 hours
Next, we calculate the population after 5 hours. Since 5 hours is 300 minutes, we plug this into the formula:
Simplifying the exponent:
So:
We know that , so:
Thus, the bacterial population after 5 hours will be approximately 6.93 billion.
Would you like to discuss the solution details or ask further questions?
Here are 5 related questions for you:
- How does the population change if the doubling period is shortened?
- What happens to the population size if the initial population is larger?
- Can you determine how many times the population doubles in a given time?
- How would you calculate the population after 10 hours?
- Can you derive a formula to predict the time when the population will reach a specific size?
Tip: Exponential growth models are widely used to describe processes where quantities grow rapidly over time, such as populations or investments.
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Math Problem Analysis
Mathematical Concepts
Exponential Growth
Doubling Time
Formulas
P(t) = P_0 * 2^(t/T)
Theorems
Exponential Growth Model
Suitable Grade Level
Grades 9-12
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