Solve.
How long will it take for the population of a certain country to double if its annual growth rate is 1.5%? Round to the nearest year. Use the exponential growth model P(t) = P0e^kt.
solve
2 = 1(e)^.015t
ln 2 = ln e^.015t
.015t = ln2
t = .693147/.015
= 46.2
Well, let's see... solving this requires a little math, but don't worry, I won't make it too complicated. The exponential growth model you provided is P(t) = P0e^kt, where P(t) represents the population at time t.
To start, let's assume the population at the beginning (t=0) is P0, and we want to find the time it takes for the population to double when the growth rate is 1.5%.
Since we're dealing with doubling the population, we can set P(t) equal to 2P0 (twice the initial population). So, we have the equation:
2P0 = P0e^(0.015t)
Now, let's solve for t. First, divide both sides by P0:
2 = e^(0.015t)
To isolate t, we need to take the natural logarithm (ln) on both sides:
ln(2) = ln(e^(0.015t))
Using the property of logarithms, ln(e^x) = x, we simplify further:
ln(2) = 0.015t
Now, divide both sides by 0.015:
t = ln(2) / 0.015
Using a calculator, we find:
t ≈ 46.53
Now, remember, we need to round to the nearest year. Since we can't have a fraction of a year, we can round it to the nearest integer:
t ≈ 47 years
So, it will take approximately 47 years for the population of the country to double with a growth rate of 1.5%. And here's a fun fact: that's a lot of birthday parties to plan!
To solve this problem, we'll use the exponential growth model P(t) = P0e^kt, where:
- P(t) represents the population at time t
- P0 represents the initial population at time t = 0
- k represents the growth rate
In this case, we want to find the time it takes for the population to double, so P(t) = 2P0.
Substituting these values into the formula, we have:
2P0 = P0e^(k*t)
To isolate t, we first divide both sides of the equation by P0:
2 = e^(k*t)
Next, we take the natural logarithm (ln) of both sides to remove the exponential:
ln(2) = ln(e^(k*t))
Using the property of logarithms, ln(e^(k*t)) simplifies to k*t:
ln(2) = k*t
Finally, to solve for t, we divide both sides of the equation by k:
t = ln(2) / k
Given that the annual growth rate is 1.5%, we can convert it to a decimal by dividing it by 100:
k = 1.5% / 100 = 0.015
Plugging this value into the equation, we have:
t = ln(2) / 0.015
Using a calculator, we can approximate this value:
t ≈ 46.17
Rounded to the nearest year, it will take approximately 46 years for the population to double.
To solve the problem using the exponential growth model, we need to find the value of t when the population P(t) is double its initial population P0.
Let's start by defining the variables in the exponential growth model:
P(t) represents the population at time t.
P0 represents the initial population.
k represents the growth rate.
In this case, we know that the annual growth rate is 1.5%. Therefore, we can set k = 0.015.
We also know that we need to find the time it takes for the population to double, which means P(t) = 2P0.
Substituting the values into the exponential growth model, we have:
2P0 = P0 * e^(0.015t)
We can simplify the equation by dividing both sides by P0:
2 = e^(0.015t)
To isolate t, we need to take the natural logarithm (ln) of both sides of the equation:
ln(2) = ln(e^(0.015t))
Using the property of logarithms (ln(e^x) = x), we have:
ln(2) = 0.015t
Finally, to solve for t, we divide both sides of the equation by 0.015:
t = ln(2) / 0.015
Using a calculator, we can compute the value of t:
t ≈ 46.53 years
Rounding to the nearest year, it will take approximately 47 years for the population of the country to double.