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D.E application

bergausstein

Active member
Jul 30, 2013
191
2. the population of a community is known to increase at the rate proportional to the number of people present at any time t. if the population is doubled in 5 years how long will it take to triple?

can help me find a model here.
 

soroban

Well-known member
Feb 2, 2012
409
Hello, bergausstein!

2. The population of a community is known to increase
at a rate proportional to the population at any time t.

We have: .[tex]\frac{dP}{dt} \:=\:kP \quad\Rightarrow\quad \frac{dP}{P} \:=\:k\,dt[/tex]

Integrate: .[tex]\ln|P| \:=\:kt+C[/tex]

. . [tex]P \:=\:e^{kt+c} \:=\:e^{kt}\cdot e^c \:=\:e^{kt}\cdot C[/tex]

Hence: .[tex]P(t) \:=\: Ce^{kt}[/tex]

When [tex]t = 0,\,P = P_o[/tex], initial population.

. . [tex]P_o \:=\:Ce^0 \quad\Rightarrow\quad C \,=\,P_o[/tex]

Therefore: .[tex]P(t) \;=\;P_oe^{kt}[/tex]



If the population is doubled in 5 years,
how long will it take to triple?

When [tex]t = 5,\;P=2\!\cdot\!P_o[/tex]

We have: .[tex]2\!\cdot\!P_o \:=\:p_oe^{5k} \quad\Rightarrow\quad e^{5k} \:=\:2 [/tex]
. . [tex]5k \:=\:\ln2 \quad\Rightarrow\quad k \:=\:\tfrac{1}{5}\ln2[/tex]
Hence: .[tex]P(t) \:=\:p_oe^{(\frac{1}{5}\ln2)t} \:=\:p_o\left(e^{\ln2}\right)^{\frac{1}{5}t}[/tex]
Then: .[tex]P(t) \;=\;P_o\!\cdot\!2^{\frac{1}{5}t}[/tex]


When will [tex]P(t) = 3\!\cdot\!P_o\,?[/tex]

.[tex]3\!\cdot\!P_o \:=\:p_o\!\cdot\!2^{\frac{1}{5}t} \quad\Rightarrow\quad 2^{\frac{1}{5}t}\:=\:3[/tex]

. . [tex]\ln\left(2^{\frac{1}{5}t}\right) \:=\:\ln(3) \quad\Rightarrow\quad \tfrac{1}{5}t\ln(2) \:=\:\ln(3) [/tex]

. . [tex]t \:=\:\frac{5\ln(3)}{\ln(2)} \:=\:7.924...[/tex]

About 7.9 years.