The half-life of a radioactive kind of yttrium is 64 hours. If you start with 568 grams of it, how much will be left after 128 hours?

Answer 1

A = 568(0.5)^128/64

A = 568(0.5)²

A = 568(0.25)

Answer 2

Let's wirte down what we know..

half life = 64 hours

starting mass = 568 grams

After each 64 hour block, our mass is halved - thus the term, half life

We can do this in a couple ways. The easiest (but doesn't ALWAYS work) is to divide our initial mass by 2 for each half life time that goes by. Therefore we start at 568 and have to have 128 hours pass.

64 hours pass and we divide by 2, getting 284. We have 64 more hours to go (128-64=64)

64 more hours pass and we divide by 2 again, getting 142. We have 0 hours left (64-64=0)

142 grams is our answer.

A second way allows us to calculate for times that aren't perfect multiples of our half life. It uses the same idea but uses exponents. We divide the time to pass (in this case 128) by the half life (64). The result is 2. Let's call this our half life count for simplicity (I'll use the letter c). That is the amount of times we have to divide our initial mass (568) by 2 to get its final weight. We have to divide by 2 twice (c=2). this is the same as saying we have to divide by 4, or 2*2 or 2². Therefore, we will divide by 2^c.

c = 128/64 = 2

final mass = initial mass / (2^c)

final mass = 568 / (2²) = 568/4 = 142

Again, we get 142 grams.

This first way is very simple and straight forward, but the second allows for you to figure out an object's mass if it goes through a lot of half lives (like 30 half lives) or if it doesn't go into an exact amount of half lives (2^1.34 is easy for the calculator, but its harder to figure out by hand how to divide by 2 1.34 times)