the population of a town grew from 20,000 to 28,000. the continuous growth rate is 15%. the equation…

the population of a town grew from 20,000 to 28,000. the continuous growth rate is 15%. the equation 20,000e^{0.15t}=28,000 represents the situation, where t is the number of years the population has been growing. about how many years has the population of the town been growing? use a calculator and round your answer to the nearest whole number.\n\no 2 years\no 9 years\no 17 years\no 22 years

the population of a town grew from 20,000 to 28,000. the continuous growth rate is 15%. the equation 20,000e^{0.15t}=28,000 represents the situation, where t is the number of years the population has been growing. about how many years has the population of the town been growing? use a calculator and round your answer to the nearest whole number.\n\no 2 years\no 9 years\no 17 years\no 22 years

Answer

Explanation:

Step1: Isolate the exponential term

Divide both sides of the equation $20000e^{0.15t}=28000$ by 20000. We get $e^{0.15t}=\frac{28000}{20000}=\frac{7}{5} = 1.4$.

Step2: Take the natural - logarithm of both sides

Since $\ln(e^{x})=x$, taking the natural - logarithm of both sides of $e^{0.15t}=1.4$ gives $\ln(e^{0.15t})=\ln(1.4)$. So, $0.15t=\ln(1.4)$.

Step3: Solve for t

Divide both sides of the equation $0.15t=\ln(1.4)$ by 0.15. We know that $\ln(1.4)\approx0.3365$, then $t=\frac{\ln(1.4)}{0.15}\approx\frac{0.3365}{0.15}\approx2.24$.

Answer:

2 years