Question

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.
2 years
9 years
17 years
22 years

Explanation:

Step1: Isolate the exponential term

Divide both sides of the equation \(20000e^{0.15t}=28000\) by 20000.
\(\frac{20000e^{0.15t}}{20000}=\frac{28000}{20000}\)
Simplify to get \(e^{0.15t} = 1.4\)

Step2: Take the natural logarithm of both sides

Take \(\ln\) of both sides: \(\ln(e^{0.15t})=\ln(1.4)\)
Using the property \(\ln(e^x)=x\), we have \(0.15t=\ln(1.4)\)

Step3: Solve for t

Divide both sides by 0.15: \(t = \frac{\ln(1.4)}{0.15}\)
Calculate \(\ln(1.4)\approx0.3365\), then \(t=\frac{0.3365}{0.15}\approx2.24\)
Round to the nearest whole number, \(t\approx2\)

Answer:

2 years