# Exponential growth law, I hardly knew thee

• Sep 16th 2010, 04:12 PM
bigdan
Exponential growth law, I hardly knew thee
Hey everybody! First time poster here, your fine forum was referred to me by another website, and here I am, ready to annoy you with endless homework problems =)

Ok, so I'm in Calc2 - recently we just learned about exponential growth laws ( P(t)=Poe^(kt) ). I get the concept, I've completed many a problem, but they threw me two curve balls on my homework:

1) An insect population triples in size after 10 months. Assuming exponential growth, when will it quadruple in size?

2) The intensity of light passing through an absorbing medium decreases exponentially with the distance traveled. Suppose the decay constant for a certain plastic block is k=3 when the distance is measured in feet. How thick must the block be to reduce the intensity by a factor of one-forth?

This homework is due on WebAssign here very shortly, and I seriously need some help if you would be so kind. Maybe the concept is just flying over my head, but I can't figure out how to relate those problems to the P(t) function. (Headbang) Anywhose, if you wouldn't mind helping out a poor college student with an answer and explanation, you would rock in my book haha
And I apologize, I know how annoying this must be, but if you could post help quickly, I would really appreciate it, as the homework is due soon

Thanks everybody!
• Sep 16th 2010, 05:22 PM
pickslides
Hi bigdan, best not to put (urgent) in the title of your thread. It usaully has the opposite desired result if you know what I mean?!

We also preffered not to be annoyed.

Anyhow

using [tex]P = P_0e^{kt}[/tex] with the information given and assuming t represents months you need to solve for k given

[tex]3 = e^{10k}[/tex] what do you get?

After you have that value, solve for t when [tex]4 = e^{kt}[/tex]
• Sep 17th 2010, 01:59 AM
HallsofIvy
By the way, $Pe^{kt}$ is a sort of "stand in" for anything of the form $Pa^t$ for positive number a.
$Pa^x= Pe^{ln(a^x)}= e^{x ln(a)}$ so $Pa^x$ is just $Pe^{kx}$ with k= ln(a).

For example, "An insect population triples in size after 10 months" means the population is multplied by 3 every 10 months. If we measure t in months, the t/10 is the number of times the population has tripled. That function can be written as $P(t)= P3^{t/10}$. Asking when it will quadruple is asking for t such that $P3^{t/10}= 4P$. Dividing both sides by P gives $3^{t/10}= 4$. Of course, you will still need a logarithm to solve that.