The population of a city is 310,000 and is increasing at a rate of 4.5% each year. Approximately when will the population reach 620,000? (Use a compound growth model.)
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To solve this problem, you can use the formula for compound growth, which is:
P = P0 * (1 + r)^t
where: - P is the future population - P0 is the initial population - r is the growth rate (expressed as a decimal) - t is the time in years
Given: P0 = 310,000 (initial population) P = 620,000 (future population we want to reach) r = 4.5% per year = 0.045 (as a decimal)
We want to find t, so we arrange the formula to solve for t:
620,000 = 310,000 * (1 + 0.045)^t
Divide both sides of the equation by 310,000 to isolate the growth factor:
2 = (1 + 0.045)^t
Now, we need to solve for t, which requires taking the logarithm of both sides of the equation. It is common to use the natural logarithm (ln) for this type of calculation:
ln(2) = t * ln(1 + 0.045)
To isolate t, divide both sides by ln(1 + 0.045):
t = ln(2) / ln(1.045)
Using a calculator, you can find the values of ln(2) and ln(1.045) and divide them to get t:
t ≈ ln(2) / ln(1.045) t ≈ 0.6931471806 / 0.04402072636 t ≈ 15.74988712
It will take approximately 15.75 years for the population to reach 620,000, given a 4.5% annual growth rate. Since we typically measure time in whole years, you can round up and say that it will take approximately 16 years for the population to double to 620,000.