This exercise uses the population growth model.
The population of the world was 7.1 billion in 2013, and the observed relative growth rate was 1.1% per year.
(a) Estimate how long it takes the population to double. (Round your answer to two decimal places.)
(b) Estimate how long it takes the population to triple. (Round your answer to two decimal places.)
The population of 7.1 billion is irrelevant in this problem
For doubling at 1.1%
(1 + .011)^n = 2
1.011^n = 2
a) your question says to estimate, and then asks for 2 decimal accuracy, which makes no sense.
anyhow.....
take log of both sides:
log (1.011^n) = log2
n log 1.011 = log 2
n = log2/log 1.011 = 63.36 years
for b, use the same steps in
1.011^t = 3
To estimate how long it takes the population to double, we can use the formula for exponential growth:
\(P(t) = P_0 \cdot e^{rt}\)
Where:
- \(P(t)\) is the population at time \(t\)
- \(P_0\) is the initial population
- \(r\) is the growth rate (expressed as a decimal)
- \(t\) is the time (in years)
In this case, \(P_0 = 7.1\) billion, \(r = 0.011\) (1.1% expressed as a decimal), and we want to find the value of \(t\) that makes \(P(t) = 2 \cdot P_0\).
(a) Let's solve this equation:
\(2 \cdot P_0 = P_0 \cdot e^{rt}\)
Divide both sides by \(P_0\) and rearrange the equation:
\(2 = e^{rt}\)
To solve for \(t\), we need to take the natural logarithm (ln) of both sides:
\(ln(2) = rt\)
Now divide both sides by \(r\) to isolate \(t\):
\(t = \frac{ln(2)}{r}\)
Substituting the values, we have:
\(t = \frac{ln(2)}{0.011}\)
Calculating this value, we find that it takes approximately 63.01 years for the population to double.
(b) Similarly, to estimate how long it takes the population to triple, we can use the same formula and find the value of \(t\) that makes \(P(t) = 3 \cdot P_0\).
Let's solve the equation:
\(3 \cdot P_0 = P_0 \cdot e^{rt}\)
Divide both sides by \(P_0\) and rearrange:
\(3 = e^{rt}\)
Take the natural logarithm of both sides:
\(ln(3) = rt\)
Divide both sides by \(r\) to isolate \(t\):
\(t = \frac{ln(3)}{r}\)
Substituting the values, we have:
\(t = \frac{ln(3)}{0.011}\)
Calculating this value, we find that it takes approximately 85.84 years for the population to triple.
To estimate how long it takes the population to double or triple using the population growth model, we can use the formula:
Time = (ln (final population / initial population)) / (ln (1 + growth rate))
Note that ln denotes the natural logarithm and the growth rate should be in decimal form. Let's calculate the answers step by step:
(a) To estimate how long it takes the population to double:
Initial population = 7.1 billion
Final population = 2 * Initial population = 14.2 billion
Growth rate = 1.1% = 0.011
Plugging these values into the formula:
Time = (ln (14.2 / 7.1)) / (ln (1 + 0.011))
To calculate this using a calculator, follow these steps:
1. Divide 14.2 by 7.1 to find the quotient.
2. Take the natural logarithm (ln) of the quotient.
3. Add 1 to the growth rate.
4. Take the natural logarithm (ln) of the result from step 3.
5. Divide the result from step 2 by the result from step 4.
6. Round the final answer to two decimal places.
(b) To estimate how long it takes the population to triple:
Initial population = 7.1 billion
Final population = 3 * Initial population = 21.3 billion
Growth rate = 1.1% = 0.011
Plugging these values into the formula:
Time = (ln (21.3 / 7.1)) / (ln (1 + 0.011))
Follow the same steps as in part (a) to calculate the final answer, rounding to two decimal places.
Please note that these estimates assume a constant growth rate and do not consider factors that may affect population growth in reality.