Question

population q has 100 members, and population r has 10,000 members. both are growing exponentially at a 5% (0.05) annual growth rate. calculate the size of each population after one year and after five years. population q after one year: 105 members population r after one year: members population q after five years: members population r after five years: 12763 members

Explanation:

Step1: Formula for exponential growth

The formula for exponential growth is \(P = P_0(1 + r)^t\), where \(P_0\) is the initial population, \(r\) is the growth rate, and \(t\) is the time in years.

Step2: Population R after one year

For population R, \(P_0=10000\), \(r = 0.05\), \(t = 1\).

$$P=10000\times(1 + 0.05)^1=10000\times1.05 = 10500$$

Step3: Population Q after five years

For population Q, \(P_0 = 100\), \(r=0.05\), \(t = 5\).

$$P=100\times(1 + 0.05)^5=100\times1.05^5$$

$$1.05^5=(1+\frac{5}{100})^5=\sum_{k = 0}^{5}\binom{5}{k}(\frac{5}{100})^k\approx1.27628$$

$$P=100\times1.27628 = 127.628\approx128$$

Answer:

Population R after one year: \(10500\) members
Population Q after five years: \(128\) members