The population of a new city is 5 million and is growing at 20% annually. How many years would it take to double at this growth rate?
3-4 years
Population Doubling Time Calculation
This problem requires us to determine how many years it will take for a city's population to double, given its initial size and a consistent annual growth rate. This is a common application of compound growth principles.
Growth Rate and Doubling Concept
We are given the following information about the city:
- Initial Population (\(P_0\)): 5 million
- Annual Growth Rate (\(r\)): 20%
The objective is to find the time (\(t\)) it takes for the population to double. If the initial population is 5 million, then the doubled population (\(P_t\)) will be \(2 \times 5 \text{ million} = 10 \text{ million}\).
The general formula for compound growth, which applies to population growth, is:
\[ P_t = P_0 (1 + r)^t \]Where:
- \(P_t\) represents the population after \(t\) years.
- \(P_0\) represents the initial population.
- \(r\) represents the annual growth rate (expressed as a decimal, so 20% becomes 0.20).
- \(t\) represents the number of years.
Annual Population Growth Calculation
To find the doubling time, we set \(P_t\) to \(2 \times P_0\):
\[ 2 \times P_0 = P_0 (1 + r)^t \]We can simplify this equation by dividing both sides by \(P_0\):
\[ 2 = (1 + r)^t \]Now, substitute the given annual growth rate \(r = 0.20\):
\[ 2 = (1 + 0.20)^t \] \[ 2 = (1.20)^t \]To find \(t\), we can calculate the value of \((1.20)^t\) for different years until it reaches or exceeds 2:
| Year (\(t\)) | Growth Factor \((1.20)^t\) | Population (\(P_0 \times (1.20)^t\)) |
|---|---|---|
| Start (Year 0) | 1.00 | 5 million |
| Year 1 | \(1.20^1 = 1.20\) | \(5 \times 1.20 = 6\) million |
| Year 2 | \(1.20^2 = 1.44\) | \(5 \times 1.44 = 7.2\) million |
| Year 3 | \(1.20^3 = 1.728\) | \(5 \times 1.728 = 8.64\) million |
| Year 4 | \(1.20^4 = 2.0736\) | \(5 \times 2.0736 = 10.368\) million |
From the calculations above:
- At the end of Year 3, the population is 8.64 million, which is still less than the doubled amount of 10 million.
- At the end of Year 4, the population is 10.368 million, which has exceeded the doubled amount of 10 million.
Therefore, the city's population will double sometime between Year 3 and Year 4.
Doubling Rule of 72 for Estimation
The Rule of 72 is a quick method to estimate the doubling time for something growing at a constant rate. It states that you can approximate the doubling time by dividing 72 by the annual growth rate percentage.
\[ \text{Doubling Time (years)} \approx \frac{72}{\text{Growth Rate Percentage}} \]Using this rule for a 20% annual growth rate:
\[ \text{Doubling Time} \approx \frac{72}{20} = 3.6 \text{ years} \]This estimation of 3.6 years further supports that the population doubling occurs within the 3-4 year range.
Doubling Years Summary
Based on the step-by-step calculation, the population of the new city growing at 20% annually would take between 3-4 years to double.
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