Suppose that the number of bacteria in a certain population increases according to an exponential growth model. A sample of 2600 bacteria selected from this population reached the size of 2873 bacteria in two and a half hours. Find the continuous growth rate per hour. Write your answer as a percentage. Do not round any intermediate computations, and round your percentage to the nearest hundredth.
I got:
2873 = 2600 e^(2.5r)
ln(2873 / 2600) = 2.5r
[ln(2873 / 2600)] / 2.5 = r
r=0.0399381
it says to round the percentage to the nearest hundreth. any help
To find the continuous growth rate per hour in an exponential growth model, we can use the formula:
N = N₀ * e^(rt)
Where:
N₀ = Initial population size
N = Final population size
r = Continuous growth rate per hour
t = Time in hours
e = Euler's number, approximately 2.71828
In this case, we are given:
Initial population size (N₀) = 2600 bacteria
Final population size (N) = 2873 bacteria
Time (t) = 2.5 hours
We can rearrange the formula to solve for r:
N/N₀ = e^(rt)
Taking the natural logarithm of both sides gives:
ln(N/N₀) = rt
Now we can substitute the given values and solve for r:
ln(2873/2600) = r * 2.5
Divide both sides by 2.5 to isolate r:
r = ln(2873/2600) / 2.5
Using a calculator, we find:
r ≈ 0.017767
To express this as a percentage, multiply by 100:
r ≈ 1.7767%
So, the continuous growth rate per hour is approximately 1.78%.