Help solve algebra question

Question

The radioactive substance krypton-85 has a decay rate of 6.3% per year. If you have 50 mg of krypton-85, how much will you have after seven years

Answer 1

There are a few ways to solve this. Following is what I consider to be the easiest.

Use this formula:

Fraction remaining = 0.5ⁿ where n = number of half lives elapsed and solve fot fraction remaining.

6.3% decays after 1yr so fraction remaining after 1 yr = 1 - 0.063 = 0.937

0.937 = 0.5ⁿ

n = 0.094 half lives have elapsed in this 1 yr. So, 1 half life (t_1/2) = 1/0.094 = 10.6 yrs

How many half lives are represented by 7 years?

7 yrs x 1 half life/10.6 yrs = 0.66 half lives

fraction remaining = 0.5^0.66 = 0.63

mass remaining = 50 mg x 0.63 = 31.6 mg

Answer 2

Hello Sam,

This problem is a classic compound exponential question, to understand it easier let's first tackle a simplified version of the problem:

Imagine you have $120, and every year you lose half of your money.

Starting out you would have $120

After one year you would have half of that, or $60

After the second year you would lose half of the $60 you have, and would end up with $30

After the third year you would lose half of the $30 you have, and would end up with $15

In this simplified example, the $120 is the equivalent of the 50mg of krypton-85 (your starting amount), and losing half (50%) is equal to the annual decay rate (6.3%).

To calculate the rate of decay from your starting point to the end of year 1, you would multiply

50mg * (1 - .063)

As (1 - .063) represents 100% - 6.3% decay, (93.7%)

From start to end of year 1: 50mg * (.937) = 46.85mg

From year 1 to year 2: 46.85mg * (.937) = 43.90mg

From year 2 to year 3: 43.90mg * (.937) = 41.13mg

And so on

This series of operations can be better represented by the following formula:

Final Quantity of Krpyton-85 After 7 Years = 50mg * (.937)⁷

This equation gives you a solution of 31.71mg