   Chapter 2.3, Problem 60E Finite Mathematics and Applied Cal...

7th Edition
Stefan Waner + 1 other
ISBN: 9781337274203

Solutions

Chapter
Section Finite Mathematics and Applied Cal...

7th Edition
Stefan Waner + 1 other
ISBN: 9781337274203
Textbook Problem

Radioactive Decay You have just isolated a new radio­active element. If you can determine its half-life, you will win the Nobel Prize in physics. You purify a sample of 2 grams. One of your colleagues steals half of it, and 3 days later, you find that 0.1 grams of the radioactive material is still left. What is the half-life? (Round your answer to three significant digits.) [HINT: First find an exponential model, then see Example 3.]

To determine

To calculate: The half life time of a radioactive element if initially it was 2 gram and despite being stolen half of initial quantity, there is still 0.1 gram left after 3 days.

Explanation

Given Information:

The initial quantity of the radioactive element was 2 gram. Half of the initial quantity is stolen. Final quantity of the radioactive element after 3 days is 0.1 gram.

Formula used:

The expression of exponential decay model is,

Q(t)=Q0ekt

Here, Q(t) is remaining amount at time t, Q0 is initial amount of the sample, k is decay constant and t is time in year.

The relation between decay constant and half-life time is,

tdk=ln2

Here, td is half-life time.

Calculation:

The initial quantity of the radioactive element is 2 gram,

The left over quantity after stolen is half of the initial quantity that is 1 gram.

Consider the expression of exponential decay model is,

Q(t)=Q0ekt

Substitute Q(t)=0.1 gram, Q0=1 gram and t=3 days in the expression Q(t)=Q0ekt.

0.1=(1)ek(3)0

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