The population of a certain country grows according to the formula
N=N0e^kt
where N is the number of people(in millions) after t years,N0 is the initial number of people(in millions) and k=1/20In 5/4.Calculate the doubling time of this population.Leave your answer in terms of In.Do not use a calculator.
please help i have no idea where to start
use the property:
ln(e^x) = x
2 = 1 e^(kt)
ln 2 = kt
t = ln2/k , replace k with the given value
To calculate the doubling time of a population, we need to find the time it takes for the population to double its initial value. In this case, the initial population is N0, and we want to find the time when the population becomes 2N0.
Let's start by substituting the given values into the equation:
N = N0 * e^(kt)
We want to find the time t when the population N is equal to 2N0:
2N0 = N0 * e^(kt)
We can cancel out N0 on both sides of the equation:
2 = e^(kt)
Next, take the natural logarithm (ln) of both sides of the equation:
ln(2) = ln(e^(kt))
The ln and e exponentials cancel each other out:
ln(2) = kt
Now, we need to solve for t, the doubling time. Rewrite the equation in terms of t:
t = ln(2) / k
Given k = (1/20)ln(5/4), substitute that into the equation:
t = ln(2) / ((1/20)ln(5/4))
To simplify the equation, divide the natural logarithms:
t = ln(2) * (20/ln(5/4))
Therefore, the doubling time of the population, in terms of ln, is ln(2) * (20/ln(5/4)).
Note: Remember to leave your answer in terms of ln, as instructed, without using a calculator.