Exponential-Decay Word Problems
Purplemath
What is the formula for exponential decay?
Exponential growth word problems work off the exponential-decay formula, A = Pekt, where A is the ending amount of whatever you're dealing with (for example, carbon-14 in a biological sample), P is the beginning amount of that same whatever, k is the decay constant, and t is time.
What is the difference between the exponential-growth formula and the exponential-decay formula?
The only difference between exponential-growth equations and exponential-decay equations is that the constant for decay situations is negative. The equation itself is just the same as for exponential growth, but you should expect a negative value for the constant. If you get a positive value, you should probably go back and check your work.
Note that the particular variables used in the equation may change from one problem to another, or from one context to another, but that the structure of the equation is always the same. For instance, all of the following represent the same relationship:
A = Pert
A = Pekt
Q = Nekt
Q = Q0ekt
No matter the particular letters used, the green variable stands for the ending amount, the blue variable stands for the beginning amount, the red variable stands for the decay constant, and the purple variable stands for time. Get comfortable with this formula; you'll be seeing a lot of it.
To solve exponential decay word problems, you may be plugging one value into the exponential-decay equation, and solving for the required result. But you may need to solve two problems in one, where you use, say, the half-life information to find the decay constant (probably by solving the exponential equation by using logarithms), and then using that value to find whatever the exercise requested.
What is an example of a radioactive-decay exercise?
- Radio-isotopes of different elements have different half-lives. Magnesium-27 has a half-life of 9.45 minutes. What is the decay constant for Magnesium-27? Round to five decimal places.
Since the decay rate is given in terms of minutes, then time t will be in minutes. However, I note that there is no beginning or ending amount given. How am I supposed to figure out what the decay constant is?
I can do this by working from the definition of "half-life": in the given amount of time (in this case, 9.45 minutes), half of the initial amount will be gone. That is, from t = 0 to t = 9.45, I will have gone from 100% of however much I'd started with (where "100%" is written as 1, being its decimal equivalent) to 50% of that amount (converted to 0.5 for use inside the formula).
Since the half-life does not depend on how much I started with, I can either pick an arbitrary beginning amount (such as 100 grams) and then calculate the decay constant after 9.45 minutes, at which point only 50 grams will remain (the other 50 grams will have mutated into some other isotope or element). Or else I can just deal with the 50% that is left. Either way, I will end up with this equation:
0.5 = e9.45k
Solving for the decay constant k, I get:
0.5 = e9.45k
ln(0.5) = 9.45k
They want me to round to five decimal places, so my answer is:
k = −0.07335
The constant was negative, as expected, because this was a decay problem. If I'd ended up with a positive value, this would have signalled to me that I'd made a mistake somewhere.
(Note: Technically, there should be units on the constant, so that the units on the time variable t will cancel off. In other words, technically, the answer should be "k = −0.07335/minute. But this probably won't matter in your math class.)
- Some people are frightened of certain medical tests because the tests involve the injection of radioactive materials. A hepatobiliary scan of my gallbladder involved an injection of 0.5 cc's (or about one-tenth of a teaspoon) of Technetium-99m, which has a half-life of almost exactly 6 hours. While undergoing the test, I heard the technician telling somebody on the phone that "in twenty-four hours, you'll be down to background radiation levels." Figure out just how much radioactive material remained from my gallbladder scan after twenty-four hours. Show your work below.
For this exercise, I need to find the ending amount A of Technetium-99m. Recalling that 1 cc (cubic centimeter) equals 1 mL (milliliter), I know that the beginning amount is P = 0.5 mL. The ending time is 24 hours. I do not have the decay constant but, by using the half-life information, I can find it. (Since this is a decay problem, I expect the constant to be negative. If I end up with a positive value, I'll know that I should go back and check my work.)
In 6 hours, there will be 50% of the original amount left, so:
0.5 = e6k
ln(0.5) = 6k
(This evaluates to about −0.1155, but I'll leave the decay constant in exact form to avoid round-off error.)
Now that I have the decay constant, I can find out how much Technitium-99m was left after twenty-four hours:
A = 0.5e(ln(0.5)/6)(24) = 0.03125
There will be no more than 0.03125 mL (or about 1/160 of a teaspoon) of Technitium-99m remaining after twenty-four hours.
It can be very useful, sometimes in unexpected places, to know that 1 cc (that is, one cubic centimeter) is equal to 1 mL (that is, one milliliter). I would recommend that you add this bit of knowledge to your store.
By the way: Technetium-99m is one of the most commonly used radioisotope for these medical purposes. Its radiation is extremely low-energy, so the chance of mutation is very low. (Whatever you're being treated for is the greater danger.) The half-life is just long enough for the doctors to have time to take their pictures. The dose I was given is about as large as these injections typically get. Your body does not easily absorb this chemical, so most of the injection is... "voided", shall we say? into the sewer system. 🚽
What is an example of solving a carbon-dating exercise?
- Carbon-14 has a half-life of 5730 years. You are presented with a document which purports to contain the recollections of a Mycenaean soldier during the Trojan War. The city of Troy was finally destroyed in about 1250 BC, or about 3250 years ago. Carbon-dating evaluates the ratio of radioactive carbon-14 to stable carbon-12. Given the amount of carbon-12 contained a measured sample cut from the document, there would have been about 1.3 × 10−12 grams of carbon-14 in the sample when the parchment was new, assuming the proposed age is correct. According to your equipment, there remains 1.0 × 10−12 grams. Is there a possibility that this is a genuine document? Or is this instead a recent forgery? Justify your conclusions.
First, as usual, I have to find the decay rate. (In real life, you'd look this up on a table, or have it programmed into your equipment, but this is math, not real life.) The half-life is 5730 years, so:
0.5 = e5730k
ln(0.5) = 5730k
I'll leave the decay constant in this "exact" form to avoid round-off error.
I have the beginning (expected) amount of C-14 and the present (ending) amount; from this information, I can calculate the age of the parchment:
Then the parchment is about 2170 years old, much less than the necessary 3250 years ago that the Trojan War took place. But the parchment is indeed old, so this isn't a total fake.
Since the parchment is genuinely old (2170 years), but clearly not old enough to be the actual writings of a soldier in the Trojan War (3250 years), either this is a much-younger copy of an earlier document (in which case it is odd that there are no references to it in other documents, since only famous works tended to be copied), or, which is more likely, this is a recent forgery written on a not-quite-old-enough ancient parchment. If possible, the ink should be tested, since a recent forgery would use recently-made ink.
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