A population of animals starts at 212,000 and grows continuously at 5% per year to 386,289. How long does it take to achieve this growth?
We can use the exponential growth formula for continuous growth:
Final amount = Initial amount * e^(rate * time)
where:
Final amount = 386,289
Initial amount = 212,000
Rate = 0.05 (5% expressed as a decimal)
Time = unknown (to be determined)
Plugging in these values into the formula, we get:
386,289 = 212,000 * e^(0.05 * time)
Divide both sides of the equation by 212,000 to isolate the exponential term:
e^(0.05 * time) = 386,289 / 212,000
Taking the natural logarithm (ln) of both sides to eliminate the exponential term:
ln(e^(0.05 * time)) = ln(386,289 / 212,000)
Simplifying the left side of the equation:
0.05 * time = ln(386,289 / 212,000)
Now, we can solve for time by dividing both sides of the equation by 0.05:
time = ln(386,289 / 212,000) / 0.05
Using a calculator, we find:
time ≈ 16.23 years
Therefore, it takes approximately 16.23 years for the population to achieve this growth.