A city's population is currently 67,900. If the population doubles every 21 years, what will the population be 84 years from now?
Understand the Problem
The question is asking us to calculate the future population of a city based on its current population and the rate at which it doubles. We need to determine how many times it will double in 84 years and then find the population at that time.
Answer
The future population will be $1,086,400$.
Answer for screen readers
The future population will be $1,086,400$.
Steps to Solve
- Calculate the number of doubling periods in 84 years
First, we need to find out how many times the population will double in 84 years. Since the population doubles every 21 years, we divide 84 by 21:
$$ \text{Number of doublings} = \frac{84}{21} = 4 $$
- Calculate the population after doublings
Next, we can find the future population by applying the doubling formula repeatedly. The formula for calculating the future population is:
$$ \text{Future Population} = \text{Current Population} \times 2^{\text{Number of doublings}} $$
In this case, the current population is 67,900, so:
$$ \text{Future Population} = 67,900 \times 2^4 $$
- Calculate the power of 2
Now we calculate $2^4$:
$$ 2^4 = 16 $$
- Calculate the final population
Finally, multiply the current population by the calculated power of 2:
$$ \text{Future Population} = 67,900 \times 16 $$
Now, let's compute that:
$$ \text{Future Population} = 1,086,400 $$
The future population will be $1,086,400$.
More Information
In this problem, we found that by doubling the population over a specific period, we can understand exponential growth. A population that doubles consistently can result in significant growth over time.
Tips
- Miscalculating the number of doublings. Ensure you divide the total years by the doubling period correctly.
- Forgetting to apply the formula for future population growth properly. Be careful with the multiplication.
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