To determine the number of years it will take for the population of the city to reach 120,000, we need to solve the given exponential growth equation for [tex]\( t \)[/tex].
The given equation for population [tex]\( P \)[/tex] in thousands is:
[tex]\[ P = 80e^{0.015t} \][/tex]
We want to find the time [tex]\( t \)[/tex] when the population [tex]\( P \)[/tex] is 120,000. Since [tex]\( P \)[/tex] is given in thousands:
[tex]\[ P = 120 \][/tex]
Thus, the equation becomes:
[tex]\[ 120 = 80e^{0.015t} \][/tex]
To isolate [tex]\( t \)[/tex], we follow these steps:
1. Divide both sides by 80:
[tex]\[ \frac{120}{80} = e^{0.015t} \][/tex]
Simplify the fraction:
[tex]\[ 1.5 = e^{0.015t} \][/tex]
2. Take the natural logarithm (ln) of both sides to eliminate the exponential function:
[tex]\[ \ln(1.5) = \ln(e^{0.015t}) \][/tex]
3. Use the property of logarithms that [tex]\( \ln(e^x) = x \)[/tex]:
[tex]\[ \ln(1.5) = 0.015t \][/tex]
4. Solve for [tex]\( t \)[/tex] by dividing both sides by 0.015:
[tex]\[ t = \frac{\ln(1.5)}{0.015} \][/tex]
5. Calculate the natural logarithm of 1.5:
[tex]\[ \ln(1.5) \approx 0.405465 \][/tex]
6. Divide by 0.015 to find [tex]\( t \)[/tex]:
[tex]\[ t = \frac{0.405465}{0.015} \approx 27.031 \][/tex]
Conclusion:
It will take approximately 27 years for the population of the city to reach 120,000.
