The population of a region is growing exponentially. There were 20 million people in 1985 (t = 0) and 62 million in 1995. Find an expression for the population at any time t, in years. Use the general exponential function and remember to use exact values.
P(t) = ? (in millions)
What population would you predict for the year 2005?
What is the doubling time in years? (Round your answer to one decimal place.)

Question

The population of a region is growing exponentially. There were 20 million people in 1985 (t = 0) and 62 million in 1995. Find an expression for the population at any time t, in years. Use the general exponential function and remember to use exact values.
P(t) = ? (in millions)
What population would you predict for the year 2005?
What is the doubling time in years? (Round your answer to one decimal place.)

anonymous · Answer

Okay, so we know it is an exponential function.  It should be in the form:$$\large P(t) = ce^{kt}$$

First we solve for $c$.  When is $e^{kt} = 1$?

anonymous · Answer

im so confused I have no idea how to solve

anonymous · Answer

Since $e^0 = 1$ then it is when $kt = 0$.
Does that make sense?

anonymous · Answer

yea I get that much

anonymous · Answer

$kt = 0$ when $k=0$ or $t=0$.  We know that $k\neq 0$.  But we do know when $t=0$.

anonymous · Answer

$$\Large P(0) = ce^{k\cdot 0} \Rightarrow 20 = ce^{k\cdot 0}$$ Can you solve for $c$ here?

anonymous · Answer

Also, do you know what I just did?

anonymous · Answer

i dont know how to solve c

anonymous · Answer

$$\Large P(0) = ce^{k\cdot 0} \Rightarrow 20 = ce^{k\cdot 0} \Rightarrow 20 = c\cdot e^0 \Rightarrow 20 = c$$ So $c = 20$.
What level of math are you in?

anonymous · Answer

Basically $c = P(0)$.  So $c$ will always be our initial amount in these exponential functions.

anonymous · Answer

Thus we have $$P(t) = 20e^{kt}$$Our next step is to solve for k.

anonymous · Answer

how do i do that

anonymous · Answer

can you please just solve it for me because im on a timed problem

anonymous · Answer

We have to use the fact that $$P(10) = 62$$We were given this when they said in 1995 the population was 1995.

anonymous · Answer

ok i got that.

anonymous · Answer

how do we get the general form to find any time

anonymous · Answer

So $$\Large 62 = 20e^{10k} \Rightarrow \frac{62}{20} = e^{10k}\Rightarrow \ln\left(\frac{62}{20}\right) = 10k \Rightarrow k = \frac{1}{10}\ln\left(\frac{62}{20}\right) $$

anonymous · Answer

whats the final answer

anonymous · Answer

It doesn't matter to me if you're timed or not.

anonymous · Answer

p(t)=20(3.1^t/10)???

anonymous · Answer

No, k = 1.131/10

anonymous · Answer

So it's $$\Large P(t) = 20e^{0.1131t}$$

anonymous · Answer

Doubling time $t_0$ such that $P(t_0) = 2P(0)$

anonymous · Answer

So solve for $t_0$ in: $$\Large 40 = 20e^{.113t_0}$$

anonymous · Answer

thank you. i got it

anonymous · Answer

how would i use that to find the doubling time.

anonymous · Answer

One variable, one equation.  Don't you know how to isolate a variable?

anonymous · Answer

of p(t)=20(3.1 ^t/10)

anonymous · Answer

i dont know the doubling time formula