Precalculus 1/19/10

Today we learned how to solve some science applications using exponential formulas such as the Law of Uninhibited Growth and Newtons Law of Cooling. Here's an example.

Example #1 Law of Uninhibited Growth

A culture of 2000 bacteria is growing in a petri dish. There are 2200 bacteria after 1 hour. Using the Law of Uninhibted growth, how many are there after 8 hours? 1day?

The Law of Uninhibited Growth

stands for the initial amount. Other values get substituted. Now solve for k


get e by itself


take natural log of both sides


solve for k. Make sure you store the variable



substitute back into original formula. Use inititial amount and t = 8

A=4287

When t = 24 (because there is 24 hours in a day)



When will there be 10,000 bacteria?


substitute 10,ooo in for A. Use the same initial value and the same k

divide both sides by 2000




Example #2 Using Newton's Law of Cooling

Here's an example that involves Newton's Law of Cooling. The formula that we'll use for this is the following:



u(t) represent the Temperature of the object after a certain time
T represent the temperature of the surrounding area (usually room temperature)
represents the initial temperature of the object

t represents the time

Just like the previous problem, we'll have to find k first, rewrite the equation, then answer the question.

Now here's an example...

An object is heated to 100 degrees. The object is 80 degrees after 5 minutes. The temperature of the room is 30? When will the temperature of the object be 50 degrees?

OK, let's find k first by substituting everything we know...

substituting...

parentheses first..

get e by itself, take the natural log of both sides


now solve for k, but store k in your calculator!


now rewrite the formula, now we are going to solve for t, with u(t)=50

solve for k


Helpful hints:

Always solve for k first, then rewrite the formula, then answer the question
You need to memorize then formula for the law of uninhibited growth, but not Newton's Law of Cooling.

You can find other examples at Mr. Corn's web page here.