Final answer:
To determine the time it takes for a population to quadruple at a 5% growth rate, we use the exponential growth equation to find that it will take approximately 27.72 years for the population to grow to four times its initial size.
Step-by-step explanation:
We are looking to calculate the time it takes for a population to grow to four times its initial size with a given relative annual growth rate of 5%. This type of problem involves exponential growth and can be solved using the formula for exponential growth, which is P = Poert, where P is the final population size, Po is the initial population size, r is the growth rate, and t is the time in years.
To find the time it takes for a population to quadruple, we set P to 4 times the initial population size Po and solve the equation for t. Using the provided annual growth rate of 5%, which is written as 0.05 in our equation, we get 4Po = Po * e^(0.05t). Simplifying this equation, we divide both sides by Po and take the natural logarithm to solve for t. The calculation works out as follows:
- Divide both sides by Po: 4 = e^(0.05t)
- Take the natural logarithm of both sides: ln(4) = 0.05t
- Solve for t: t = ln(4)/0.05
By calculating this, we find that t is approximately 27.72 years. Therefore, it will take about 27.72 years for the population to become four times its initial value with a relative growth rate of 5% per year.