intro
2 short movies = fission/natural radioactivity
Part 1 : NATURAL RADIOACTIVITY
REVIEW MATH
Exponential growth means each time a x (indepdendent variable like time) increases by 1 (like 1 year, 1 day ..) , y increases by a non constant amount =
constant percentage of the previous y. Like money in an account with a 5% interest. Say at t=0 yo = $100.
After 1 year, y =100 x 1.05 (increase by 5%), after 2 years y = 100x1.05 x 1.05 , after 3 years y=100 x 1.05x1.05x1.05 x....
So after t years you have in your account : y(t) = 100( 1 + 0.05 )t This function models the exponential growth of your money
in your account. You could also write y = yo (1.05)t = yo e ln(1.05)t e is the natural exponential.
The 2 functions model the same growth but it is easier to use the natural exponential functions.
The general formula for exponential growth is : y(t) = yo( 1 + r )t with r the growth %rate , yo the value at t=0 and t the number of years.
or y = yo e ln(1+r)t
We can also model an exponential decay using exponential.
y = yo (1- r ) t r tells you by how much y decreases every time x increases by one. In %. r is the decay %rate.
If you get a 20% discount, it means you are only going to pay 80% of the price (so 1 - 0.2 = .8 and y = yo(0.8)t
If a population of bacteria decreases by 5% every day we have y = yo(0.8)t or y = yo eln(0.8) t
SOME PROBLEMS :
1) exponential growth
The 1986 explosion of Chernobyl nuclear power plant in the former Soviet Union sent about 1000kg of radioactive cesium-137
into the atmosphere. The function f(x)= 1000 (0.5) x/30 describes the amount in Chernobyl x years after 1986.
(so the mass of cesium decreases by 50% every 30 years, it halves every 30 years. We say the half life of cesium 137 is 30 years).
If even 100kg of cesium -137 remain in Chernobyl's atmosphere, the area is considered unsafe for human habitation.
Find f(80) and determine if Chernobyl will be safe for human habitation by 2066.
Carbon dating is used to determine the age of fossils and artifacts. The method is based on considering the percentage of carbon-14 remaining
in the fossils. Carbon-14 decay exponentially with a half-life of approximately 5715 years. The half-life of a substance is the time
required for half of a sample to disintegrate. Thus, after 5715 years a given amount of carbon-14 will have decayed
to half the original amount. Carbon dating is useful for artifacts and fossils up to 80,000 years old.
Older objects do not have enough carbon-14 to determine age accurately.
The model is : y = yo (1-0.5) t/to to is the half-life. or y=yo e ln(0.5)t/to
2) Carbon-14 the Dead sea Scrolls.
A) Use the fact that after 5715 years a given amount of carbon-14 years will have decayed to half the original amount to find the exponential
decay model for carbon-14.
hint: y=yo(1-r) t/5715 with r = 0.5 convert to a natural exponential function
3) In 1947 , earthenware jars containing what are known as the Dead Sea Scrolls were found by an Arab Bedouin herdsman.
Analysis indicated that the scroll wrapping contained 76% of their original carbon-14. Estimate the age of the Dead Sea Scrolls.
(hint you should find 2,263 years about between the time it was a " cow: and 1947 )
4) Strontium-90 is a waste product from nuclear reactor . As a consequence of fallout from atmospheric nuclear tests, we all
have a measurable amount of strontium-90 in our bones.
A) The half-life is 28 years. After 28 years a given amount of strontium will have decayed to half the original amount.
Find the exponential decay model for strontium-90.
B) Suppose that a nuclear accident occurs and releases 60 grams of strontium-90 into the atmosphere. How long will it take
for strontium to decay to a level of 10 grams ?
(hint answer is 72 years).
5) The AUgust 1978 issue of National Geographic described the 1964 find of bones of a newly discovered dinosaur
weighing 170 pounds, measuring 9 feet, with a 6-inch claw on one toe of each hind foot. The age of the dinosaur was estimated using
potassium 40 dating of rocks around the bones.
A) Potassium-40 decay exponentially with a half life of 1.31 billion years. Use the fact that after 1.31 billion years
a given amount of potassium40 will have decayed to half the original amount to show that the decay model
for potassium40 is given by A = Ao e-0.52912t , t in billion years.
B) Analysis show that the rocks surrounding the bones indicated that 94.5% of the original amount of potassium was
still present. A = 0.945Ao. Estimate the age of the dinsaur.
6) Fill the table:
| radioactive susbtance | Half-life= to | decay rate = k in Y=Yo e kt k=ln(0.5)/ to |
| Tritium | 5.5% per year so k = -0.055 | |
| krypton-85 | 6.3% per year = - 0.063 | |
| Radium 226 | 1620 years | |
| Uranium 4560 | 4560 years |