The population of a town with a 2016 population of 105,000 grows at a rate of $2.6\mathrm{\%}$ per year.
a. Find the rate constant $k$ and use it to devise an exponential growth function that fits the given data.
b. In what year will the population reach 175,000 ?
a. Find the rate constant $k$.
$\mathrm{k}=\ln (1.026)$
(Type an exact answer.)
Devise an exponential growth function that fits the given data. Use the value of $\mathrm{k}$ found in the previous step.
$y(t)=$
(Type an expression using $t$ as the variable. Type an exact answer.)

Step 1 ：The general form of an exponential growth function is $y(t)=y_0\ast e^{kt}$, where $y_0$ is the initial amount, $k$ is the rate constant, and $t$ is the time. In this case, the initial population is 105,000 and the growth rate is 2.6% per year.

Step 2 ：We can convert the percentage to a decimal and use it to find the rate constant $k$ by taking the natural logarithm of 1 plus the growth rate. So, $k=\ln (1.026)$, which is approximately 0.02567.

Step 3 ：Substitute the values into the exponential growth function, we get $y(t)=105000\ast e^{0.02567t}$.

Step 4 ：To find out in what year will the population reach 175,000, we set $y(t)=175000$ and solve for $t$.

Step 5 ：By solving the equation, we get $t$ is approximately 19.901459554267042. Since the initial year is 2016, we add $t$ to 2016 and get the year is approximately 2035.9014595542671.

Step 6 ：Rounding up to the nearest whole year, the population will reach 175,000 in the year 2036.

Step 7 ：So, the final answers are $k\approx \boxed{{\displaystyle 0.02567}}$, $y(t)=105000\ast e^{0.02567t}$, and the year is $\boxed{{\displaystyle 2036}}$.