Answer to Question #211917 in Calculus for Faith

Question #211917

The population of a certain community is known to increase at a rate proportional to the number of people present any time.if the population had double in 5 years,how long will it take to triple?


1
Expert's answer
2021-07-05T04:15:01-0400

Let "n" be the number of people present any time. Then "y'=ky," and hence "\\frac{dy}{y}=kdt." It follows that "\\int\\frac{dy}{y}=\\int kdt," and thus "\\ln|y|=kt+C," that is "y(t)=Ce^{kt}." It follows that "y(5)=2y(0)," that is "Ce^{5k}=2C." We conclude that "k=\\frac{1}{5}\\ln 2," and consequently, "y(t)=Ce^{\\frac{1}{5}(\\ln 2)t}=C2^{\\frac{t}{5}}." Let us find when the population will triple. It follows that "y(t)=3y(0)," that is "C2^{\\frac{t}{5}}=3C." We have that "t=5\\log_2 3\\approx 7.9"


Answer: 7.9 years


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