Growth problem? Any help would be greatly appreciated!

**camjenson** · November 11th 2012, 08:06 PM

thanks!

**Soroban** · November 11th 2012, 09:01 PM

Hello, camjenson!

A colony of people grows at a rate directly proportional to the size of the population.
The colony triples every two hours.

1. (a) Write differential equation that describes the colony's growth.
. . (b) Find the general solution.

$(a)\;\frac{dP}{dt} \:=\:k\!\cdot\! P$

(b) We have: . $\frac{dP}{P} \:=\:k\,dt$

Integrate: . $\ln P \:=\:kt + c \quad\Rightarrow\quad P \:=\:e^{kt+c} \:=\:e^{kt}\!\cdot\! e^c} \:=\:e^{kt}\cdot C$

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

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

So we have: . $P_o \:=\:Ce^{0} \quad\Rightarrow\quad C \,=\,P_o$

Therefore, the equation is: . $P \;=\;P_oe^{kt}$

2. What's the value of the constant of proportionality?

The population triples every two hours.
When $t = 2,\;P = 3P_o$

We have: . $P_oe^{2k} \:=\:3P_o \quad\Rightarrow\quad e^{2k} \,=\,3 \quad\Rightarrow\quad 2k \,=\,\ln3$

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

3. At t = 9 hours, the population is 800 million, what is the initial population?

The equation is: . $P \;=\;P_o\,e^{\frac{1}{2}\ln(3)\cdot t} \;=\;P_o\left(e^{\ln 3}\right)^{\frac{1}{2}t}$

And we have: . $P \;=\;P_o\!\cdot\!3^{\frac{1}{2}t}$

When $t = 9,\:P = 800,\!000,\!000.$

. . $P_o\!\cdot\!3^{4.5} \:=\:800,\!000,\!000$

. . $P_o \;=\;\frac{800,\!000,\!000}{3^{4.5}} \;\approx\; 5,\!702,\!225$

Thread: Growth problem? Any help would be greatly appreciated!

LinkBack

Thread Tools

Display

Growth problem? Any help would be greatly appreciated!

Re: Growth problem? Any help would be greatly appreciated!

Similar Math Help Forum Discussions

Problem Solve Equation (Any Help - Greatly Appreciated)

Any help would be greatly appreciated!

Word problem based on Empirical Rule, Any help would greatly be appreciated

Can Anyone Figure Out This Problem(Greatly Appreciated!!)

Compound Interest Problem! Any help greatly appreciated. Challenge

Search Tags