Calculate Exponential Bacterial Growth: Doubling Time and Population Increase
Understanding the growth patterns of bacterial cultures is crucial in various scientific and medical applications. Exponential growth models help us predict and analyze the population dynamics of such cultures. In this article, we will solve a specific problem where the number of bacteria in a culture is doubling according to a given equation. We will walk through the steps to determine the time required for the bacterial population to increase from 100,000 to 25,600,000.
Understanding the Model
The number of bacteria, ( N_t ), in a culture at any given time ( t ) is modeled as an exponential growth function:
[ N_t N_0 cdot 2^{frac{t}{7}} ]
Where: ( N_t ) is the number of bacteria at time ( t ). ( N_0 ) is the initial number of bacteria when the experiment begins. ( t ) is the time in hours.
Problem Statement
Given the initial number of bacteria ( N_0 100,000 ) and the target number of bacteria ( N_t 25,600,000 ), find the number of hours later when the number of bacteria will reach ( N_t ).
Solving the Problem
To solve the problem, we start with the given equation:
[ N_t N_0 cdot 2^{frac{t}{7}} ]
Substituting the known values into the equation:
[ 25,600,000 100,000 cdot 2^{frac{t}{7}} ]
Dividing both sides by 100,000:
[ frac{25600000}{100000} 2^{frac{t}{7}} ]
Simplifying:
[ 256 2^{frac{t}{7}} ]
Expressing 256 as a power of 2:
[ 256 2^8 ]
Therefore, we can rewrite the equation as:
[ 2^8 2^{frac{t}{7}} ]
Solving for ( t ) by setting the exponents equal:
[ 8 frac{t}{7} ]
Solving for ( t ) by multiplying both sides by 7:
[ t 8 times 7 56 , text{hours} ]
Thus, the number of bacteria will reach 25,600,000 after 56 hours.
Relating to the Original Problem Statement
The problem suggests filling in the time value and doubling time. You can set any value for time. Here's a breakdown of the solution:
[ 25600000 100000 times 2^{frac{t}{7}} ]
[ 2^{frac{t}{7}} 256 ]
[ ln(2^{frac{t}{7}}) ln(256) ]
[ frac{t}{7} ln(2) ln(256) ]
[ frac{t}{7} 8 ]
[ t 56 , text{hours} ]
Conclusion
Using the exponential growth equation for bacterial cultures, we determined that it takes 56 hours for the bacterial population to grow from 100,000 to 25,600,000. This problem demonstrates the practical application of exponential growth models in scientific analysis.