Hi braggy,
Because 36 days is an integer multiple of the half-life, we can work
out the answer by taking one half-life period at a time:
At day 0, we have 28 grams
At day 12, we have half of this (14 grams)
At day 24, we have half again (7 grams)
At day 36, we have half again (3.5 grams)
Alternatively, we can use a general formula for exponential decay
which will allow us to work out the amount of substance remaining
after any number of days.
The formula for exponential decay is
s = s0 * e ^ (k * t)
where
s is the amount of the substance at any time t
s0 is the amount of the substance at day 0
e is the base of the natural logarithms (2.718281828459...)
k is some constant showing how quickly the substance decays
t is the time since day 0
* represents multiplication
^ represents exponentiation
First we must find the value of 'k' for this substance. What we do
know is that after 12 days, half of the substance remains (14 grams).
We plug these values into the formula:
14 = 28 * e ^ (k * 12)
Divide both sides by 28:
0.5 = e ^ (k * 12)
Take the natural logarithm of both sides:
ln 0.5 = ln(e ^ (k * 12))
Cancel the 'ln' and the exponentiation on the right-hand-side, which
we can do because e^(ln x)=x for any x:
ln 0.5 = k * 12
Solve for 'k':
k = (ln 0.5)/12 = -0.057762265046662
Now, having found 'k', we can use our original formula to find the
amount of substance remaining after 36 days:
s = 28 * e ^ (-0.057762265046662 * 36)
= 3.5 grams
A worked example in more detail can be found here:
http://math.usask.ca/emr/examples/expdeceg.html
If you don't understand any part of my explanation above, please ask
for more details by requesting a clarification.
Google Search Strategy:
"exponential decay" formula
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Regards,
eiffel-ga |