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: 29

Answer

After around 19 years, the population of Knoxville would reach 1 million.

Work Step by Step

Here is what we have: - The population of Knoxville currently. - The annual increase rate. So to calculate the population of Knoxville 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 Knoxville currently $(p_n=500.000)$ $r$: annual increase rate $(r=3.75\%=0.0375)$ $t$: the amount of time which has passed from now until the estimated time in the future (years) $p$: the population of Knoxville after time $t$ that we are estimating Here, as you can see, we need to find when the population would reach 1 million, which in essence is to find $t$ so that $p=1.000.000$ To find $t$, we substitute the known values: $$500000(1+0.0375)^t=1000000$$ $$500000\times1.0375^t=1000000$$ $$1.0375^t=2$$ Here, we use graphing calculator and find out that $t\approx18.828\approx19$ (years) Therefore, we can conclude that after around 19 years, the population of Knoxville would reach 1 million.
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