Questions: Country A has an exponential growth rate of 3.8% per year. The population is currently 4,891,000, 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?

Country A has an exponential growth rate of 3.8% per year. The population is currently 4,891,000, 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?

Transcript text: Country A has an exponential growth rate of $3.8 \%$ per year. The population is currently $4,891,000$, 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?

Solution

Solution Steps

Step 1: Convert the annual growth rate from a percentage to a decimal

To convert the growth rate to a decimal, we divide the percentage by 100: $r = \frac{3.8}{100} = 0.038$.

Step 2: Use the formula for exponential growth

The formula for exponential growth is $P = P_0 \times (1 + r)^t$, where $P$ is the population at time $t$, $P_0$ is the initial population, $r$ is the growth rate per period, and $t$ is the number of periods (years).

Step 3: Solve for $t$ when $P = A$

Substituting $A$ for $P$ and solving for $t$, we get $t = \frac{\log(A/P_0)}{\log(1 + r)} = 222.98$ years.