Expert: Paul Klarreich - 6/18/2009

Question
I'm having trouble with one of my homework word problems.

Problem:

The amount of a radioactive element in a lab. sample has decreased 60 % compared to what it was 5 days ago. What is the half-life for this element? What is the percent decay rate? What is the continuousness decay rate?

Answer
Questioner:   Denise
Country:  United States
Category:  Advanced Math
Private:  No
 
Subject:  PreCalculus
Question:  I'm having trouble with one of my homework word problems.

Problem:

The amount of a radioactive element in a lab. sample has decreased 60 % compared to what it was 5 days ago. What is the half-life for this element? What is the percent decay rate? What is the continuousness decay rate?
..............................................
Hi, Denise,

You will use the basic growth-and-decay equation:

A = A0 exp(-kt), where:

A is the amount at any t.
A0 is the amount at t = 0.
k is a constant to be computed.

Your half-life will be the value of t when  A = A0/2.

For the sentence: "The amount ... has decreased 60 %", use what you learned in pre-pre-pre-pre-pre-calculus (i.e. the 8th grade):

A = 0.40 A0

Write:

0.40 A0 = A0 exp(-k(5 days), and solve for k:

0.40 = exp(-5k)
-5k = ln(0.40)
  k =  - ln(0.40)/5

Use your calculator.

Mine gives:  

k = 0.1832581463748310130367054423536, or so.

Now for your half-life, try solving:

0.5 A0 = A0 exp(-kt)

for t, where you now know k. (Use your calculator again.)

0.5  =  exp(-kt)

-kt = ln(0.5)

t = ln(0.5)/(-k)

I get  3.7823539868301501471605268048024 days, approx.

As to the other questions:

What is the percent decay rate? What is the continuousness decay rate?

Sorry, but you must define those terms first.