# D.E application

#### bergausstein

##### Active member
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
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: .$$\frac{dP}{dt} \:=\:kP \quad\Rightarrow\quad \frac{dP}{P} \:=\:k\,dt$$

Integrate: .$$\ln|P| \:=\:kt+C$$

. . $$P \:=\:e^{kt+c} \:=\:e^{kt}\cdot e^c \:=\:e^{kt}\cdot C$$

Hence: .$$P(t) \:=\: Ce^{kt}$$

When $$t = 0,\,P = P_o$$, initial population.

. . $$P_o \:=\:Ce^0 \quad\Rightarrow\quad C \,=\,P_o$$

Therefore: .$$P(t) \;=\;P_oe^{kt}$$

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

When $$t = 5,\;P=2\!\cdot\!P_o$$

We have: .$$2\!\cdot\!P_o \:=\ _oe^{5k} \quad\Rightarrow\quad e^{5k} \:=\:2$$
. . $$5k \:=\:\ln2 \quad\Rightarrow\quad k \:=\:\tfrac{1}{5}\ln2$$
Hence: .$$P(t) \:=\ _oe^{(\frac{1}{5}\ln2)t} \:=\ _o\left(e^{\ln2}\right)^{\frac{1}{5}t}$$
Then: .$$P(t) \;=\;P_o\!\cdot\!2^{\frac{1}{5}t}$$

When will $$P(t) = 3\!\cdot\!P_o\,?$$

.$$3\!\cdot\!P_o \:=\ _o\!\cdot\!2^{\frac{1}{5}t} \quad\Rightarrow\quad 2^{\frac{1}{5}t}\:=\:3$$

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

. . $$t \:=\:\frac{5\ln(3)}{\ln(2)} \:=\:7.924...$$