The population of some countries has a relative growth rate of 2.75% per year. At this rate, how many years (to the nearest tenth of a year) will it take the population to double?
2 = 1.0275^n
log 2 = n log 1.0275
n = 25.6 years
To find out how many years it will take for a population to double with a given relative growth rate, we can use the formula for exponential growth:
Population = Initial Population × (1 + Growth Rate)^Time
In this case, the relative growth rate is 2.75%, which can be expressed as a decimal by dividing it by 100: 2.75% / 100 = 0.0275.
Let's denote the time it takes for the population to double as "t". Therefore, the population at time "t" will be twice the initial population:
2 × Initial Population = Initial Population × (1 + Growth Rate)^t
To solve for "t", we need to isolate it on one side of the equation. We can divide both sides by the initial population:
2 = (1 + Growth Rate)^t
Now, to solve for "t", we can take the logarithm (base 10 or natural logarithm) of both sides of the equation. Let's use the natural logarithm (ln):
ln(2) = ln[(1 + Growth Rate)^t]
ln(2) = t × ln(1 + Growth Rate)
Finally, to find the value of "t", we divide both sides of the equation by ln(1 + Growth Rate):
t = ln(2) / ln(1 + Growth Rate)
Substituting the value of Growth Rate (0.0275) into the equation:
t = ln(2) / ln(1 + 0.0275)
Using a calculator, we can calculate the approximate value of "t". Plugging these values into the equation, we find:
t ≈ ln(2) / ln(1.0275) ≈ 25.2
Therefore, it will take approximately 25.2 years (to the nearest tenth of a year) for the population to double with a relative growth rate of 2.75% per year.