University Calculus: Early Transcendentals (3rd Edition)

Published by Pearson
ISBN 10: 0321999584
ISBN 13: 978-0-32199-958-0

Chapter 1 - Section 1.5 - Exponential Functions - Exercises - Page 38: 30

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.
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