Let's solve this problem step-by-step.
### Part A:
We start with the given population growth formula:
[tex]\[ P = 5000 e^{0.0238 t} \][/tex]
We want to find the number of years [tex]\( t \)[/tex] it will take for the population to reach 30,000. So, we set [tex]\( P = 30000 \)[/tex]:
[tex]\[ 30000 = 5000 e^{0.0238 t} \][/tex]
First, we divide both sides of the equation by 5000 to isolate the exponential term:
[tex]\[ \frac{30000}{5000} = e^{0.0238 t} \][/tex]
[tex]\[ 6 = e^{0.0238 t} \][/tex]
Next, we take the natural logarithm (ln) of both sides to solve for [tex]\( t \)[/tex]:
[tex]\[ \ln(6) = \ln(e^{0.0238 t}) \][/tex]
Using the logarithm property [tex]\( \ln(e^x) = x \)[/tex], we get:
[tex]\[ \ln(6) = 0.0238 t \][/tex]
Finally, we solve for [tex]\( t \)[/tex]:
[tex]\[ t = \frac{\ln(6)}{0.0238} \][/tex]
So the equation that represents the number of years [tex]\( t \)[/tex] is:
[tex]\[ t = \frac{\ln(6)}{0.0238} \][/tex]
### Part B:
To find the approximate number of years, we evaluate the equation [tex]\( t = \frac{\ln(6)}{0.0238} \)[/tex].
The numerical result of this expression is approximately 75 years when rounded to the nearest whole number.
Therefore, it will take approximately:
[tex]\[ 75 \text{ years} \][/tex]
for the city's population to reach 30,000.
