The Grand and Glorious Exponential Growth Page!
So as promise, here are some supplementary notes.
Ready?
Okay! Here we go!
**** Notation Note ****
P(0) would be written in your books as “P sub zero”
e ^ rt means “e raised to the rt power” so whenever I use this symbol: “^” it means a power.
I’m trying to use similar notation to how you would enter things on a calculator
The Important Formulas!
P(t) = P(0) e ^rt
P'(t) = r * P(t) How do we get this again?
P(t) = the amount that we have after any given time, t
P(0) = the intial amount
r= growth constant
t = time
e = e (The number)
P'(t) = how fast the population is growing.
The Problem!
Let P(t) be the population (In millions) of a certain city t years after 1990 and suppose that P(t) satisfies the differential equation:
P'(t) = .02 * P(t)
P(0) = 3.
A) Find the Formula for P(t).
This is something we should do anyway, even if the problem doesn’t ask for it directly.
The formula for P(t) is one of the Important Formulas
The formula, once again, is P(t) = P(0)* e^rt.
Now we need to fill in all the blanks we can.
Our P(0) is given to us in the problem to be equal to 3.
Our r is also given to us in the problem to be .02 because the other Important Formula
is P'(t) = r * P(t).
So, lo and behold, our formula that we want is:
P(t) = 3 * e^.02t
B) What was the intial population, that is, the population in 1990?
We sorta found this already in part A
Inital Population = P(0) = 3 million.
C) What is the growth constant?
Again, we found this already
Growth Constant = r = .02
D) What was the population in 1998?
1998 is 8 years after 1990
(which is our starting date according to the problem)
This means that:
t = 8
So we can plug 8 in to P(t) for t
P(8) = 3 * e .02(8)
= 3.52 million Calculator work!
E) Use the differential equation to determine how fast the population is growing when it reaches 4 million people.
Differential Equation means we use the P'(t) equation.
“When the population reaches 4 million” This means when we have 4 million people.
Remember that P(t) means, how much we have.
So we are going to use the equation: P'(t) = .02 P(t)
AND P(t) (how much we have) = 4 So…
P'(t) = .02 * 4 = .08 which is .08 million people per year, or 80,000
F) How large is the population when it is growing at the rate of 70,000 people per year?
Recall, P'(t) means growth rate. So…
70,000 = .02 * P(t)
Dividing both sides by .02 gives:
P(t) = 3.5 million.
We do NOT need to solve for t because they did not ask us anything about time.
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To Sum Up
The best thing to do is write out the two Important Equations
and then fill in all of the variables.
This is the best advice that I can give.
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How do we find P'(t) = r * P(t) again?
Well, since you asked…
P(t) = P(0)e^rt
This is the original formula we started with
P'(t) = (P(0)e^rt) * r
Here I took the derivative.
If you recall, the rule for doing “e”‘s is to rewrite the original statement,
so I re-wrote e^rt and then multiply by the derivative of the power.
So the derivative of “rt” is simply “r”
P'(t) = r * P(t)
Since P(0)e^rt = P(t)
I can replace P(0)e^rt by P(t), giving me the equation.