Advanced Algebra - Exponential decay

1. Google “Chernobyl Disaster” to learn about
what it was and identify some key points that
you learn.

3. The byproduct of nuclear power are the
radioactive isotopes that get produced.
If not properly contained, exposure to these
isotopes produces unwanted effects to the
body(depending n how much exposure and for
how long.
Even though there was a couple of isotopes in the
Chernobyl disaster, to keep things simple we will
focus on Plutonium 238 (PU-238)

4. Objective:
1) Calculate exponential decay by applying it
to the half life of an element
2) Write an exponential decay function given
two consecutive data points.

5. The Exponential decay function has the form 𝑓 𝑥 = 𝑏 𝑥
where 0 > |b| > 1
𝑓 𝑥 = 2
1
2
𝑥

6. Half-life is the amount of time it takes an element to decay.
The equation for half-life is:
𝑓 𝑡 = 𝑎(0.5) 𝑡
 a = original amount
 t = time
During Chernobyl let’s say:
 23.7 g of Pu-238 were released
 The half life of Pu-238 is 88 years
 0.00000412 g is deadly to humans
It has been 30 years since Chernobyl, how much Pu-238 is
left?

7. Step 1) Calculate the number of half-lives there were so far:
𝐶𝑢𝑟𝑟𝑒𝑛𝑡 𝑎𝑚𝑜𝑢𝑛𝑡 𝑜𝑓 𝑡𝑖𝑚𝑒
𝐻𝑎𝑙𝑓 𝑙𝑖𝑓𝑒
=
30
88
= 0.3409
Step 2) Solve for amount using your answer from #1 as t:
23.7 0.5 0.3409 = 18.7123 𝑔𝑟𝑎𝑚𝑠 is left
You try:
Will the immediate area around Chernobyl be safe
in 10,000 year? Show calculation to justify your
answer.

8. An explosion releases 2 grams of Cesium-137 (another
harmful isotope) with a half life of 30 years. How much
Cesium-137 will be left after 50 year?

9. Suppose the rainforest is getting cut down at an
exponential rate in a certain country.
There were 30,000 acres of rain forest last year. Now there
are 28,600 acres left.
1) Write an exponential decay function to model the
amount of rainforest
2) Use your function to find out how much acres will be
left in 30 years.

10. To write an exponential decay function:
1)Take the current amount and make that year zero to write
the ordered pair (0, 28,600)
-Your original amount is a=28,600
2) Take the amount from the year before and write it as the
ordered pair (-1,30,000) . Set the y-value from the previous
year equal to
𝑎
𝑏
and solve for b since you know a.
28,600
𝑏
= 30,000
Solving, we get: b=0.953
3) Write your function: 𝑓 𝑡 = 28,600 .953 𝑡

11. So, after 30 years:
𝑓 30 = 28,600 .953 30
6748 Acres left after 30 years

12. The population of a city is shrinking exponentially. Last
year the population was 200,000 people. This year there
are 194,750 people left.
1) Write a function to model the population of the city
2) Use your function to calculate how much people will be
left in 20 years
3) About when will the population hit 50,000 people?