#### Answer

(a) The population in 1915 is 12315 and in 1940 is 24265.
(b) In year 1967, the population is estimated to reach 50,000.

#### Work Step by Step

Here is what we have:
- The population of Silver Run in 1890.
- The annual increase rate.
So to calculate the population of Silver Run at a given time in the future, we would design an exponential model like this:
$$p = p_n(1+r)^t$$
$p_n$: the population of Silver Run in 1890 $(p_n=6250)$
$r$: annual increase rate $(r=2.75\%=0.0275)$
$t$: the amount of time which has passed from 1890 till the estimated time (years)
$p$: the population of Silver Run after time $t$
(a) We need to estimate the population in 1915 and 1940.
- In 1915, which is 25 years from 1890, so $t=25$.
$$p=6250\times(1+0.0275)^{25}\approx12315$$
- In 1940, which is 50 years from 1890, so $t=50$.
$$p=6250\times(1+0.0275)^{50}\approx24265$$
(b) We need to find when the population would reach 50,000, which in essence is to find $t$ so that $p=50000$
To find $t$, we substitute the known values:
$$6250(1+0.0275)^t=50000$$
$$1.0275^t=8$$
Here, we use graphing calculator and find out that $t\approx76.651\approx77$ (years), which is the year $1890+77=1967$
Therefore, we can conclude that in year 1967, the population of Silver Run would reach 50,000.