Question 10: In the year \(2016\), the population of city was \(22\) millions and was growing at a rate of \(2.5\%\) per year. The function \(p(t)=22(1.025)^t\) gives the population in millions, \(t\) years after \(2016\). Use the model to determine in which year the poploluation will reach \(35\) millions. Round the answer to the nearest year.
Solution:
The funciton is given that
\[p=22( 1.025)^{t}\]
Taking logarithm on both sides, we have
\[
\log p=\log\left( 22( 1.025)^{t}\right)
\]
Put I \(p=35\) in above function, we have
\[
\begin{align*}
\log 35&=\log\left( 22( 1.025)^{t}\right)\\
\log 35&=\log 22+\log( 1.025)^{t}\\
1.5441&=1.3424+t( 0.0107)\\
t( 0.0107)&=1.5441-1.3424\\
t( 0.0107) &=0.2017\\
t&=\frac{0.2017}{0.0107}\\
t&=18.8\approx 19\text{ Years}
\end{align*}
\]
The population will reach \(35\) million in \(2016+19=2035\).