University Calculus: Early Transcendentals (3rd Edition)

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.