Question: Country A has an exponential growth rate of 2.7% per year. The population is currently 5,769,000,…

Country A has an exponential growth rate of 2.7% per year. The population is currently 5,769,000, and the land area of Country A is 20,000,000,000 square yards. Assuming this growth rate continues and is exponential, after how long will there be one person for every square yard of land?

This will happen in [ ] year(s). (Round to the nearest integer.)

Solution

To solve this problem, we use the formula for exponential growth: \[ P(t) = P_0 \times (1 + r)^t \] where: - \( P(t) \) is the future population, - \( P_0 = 5,769,000 \) is the initial population, - \( r = 0.027 \) (2.7% growth rate), - \( t \) is the time in years. We want to find \( t \) when the population equals the land area, which is \( 20,000,000,000 \). Set \( P(t) = 20,000,000,000 \): \[ 20,000,000,000 = 5,769,000 \times (1 + 0.027)^t \] Divide both sides by 5,769,000: \[ \frac{20,000,000,000}{5,769,000} = (1.027)^t \] Calculate the left side: \[ \frac{20,000,000,000}{5,769,000} \approx 3467.20 \] So: \[ 3467.20 = (1.027)^t \] Take the logarithm of both sides: \[ \log(3467.20) = t \times \log(1.027) \] Solve for \( t \): \[ t = \frac{\log(3467.20)}{\log(1.027)} \] Calculate \( t \): \[ t \approx \frac{3.540}{0.0115} \approx 307.83 \] Round to the nearest integer: The number of years is approximately \( 308 \).