SOLUTION: There are 3 parts to this questions. I have been stuck on this question for a long time. PLEASE HELP!! A storage tank contains a radioactive element with a half-life of 6500 yea

Question 845641: There are 3 parts to this questions. I have been stuck on this question for a long time. PLEASE HELP!!
A storage tank contains a radioactive element with a half-life of 6500 years. Let f(t) be the percentage of the element that remains at t years since it was placed in the tank.
Question A: Find an equation of f.
Question B: What percentage of the element will remain after 100 years?
Question C: What percentage of the element remains after 26,000 years? [NOTE: this result could be found using the equation of f, but we'll use a different approach to find the percentage].
PLEASE HELP ME SOLVE THIS PROBLEM!!! and PLEASE SHOW WORK SO I CAN UNDERSTAND IT!
Thanks!!
Answer by KMST(5328) (Show Source): You can put this solution on YOUR website!
A After one half-life the fraction of the element that remains is .
After half-lives the fraction of the element that remains is of , which is .
After half-lives the fraction of the element that remains is .
The same idea works for any number of half-lives, even if that number is not an integer.
years is half-lives,
After that time, the fraction of the element that remains is

As a percentage, it is .
That is an exponential function with base , and that is not a fashionable base.
Calculators have exponential functions with base and with base .
The most popular base for exponential functions is the irrational number .
The reason for that is that calculus works better with as a base,
so even if you do not intend to ever study calculus, they make you use rather than .
It is customary to write the function as
or .
It is really the same thing, because the natural logarithm of is
.
People memorize the "formula" for the exponential function with base without understanding it.
I never memorized that formula, but I can deduce the one with as a base from the definition of half-life, and then I can "translate" it to base .

B When , and

So after 100 years 98.9% of the radioactive element remains.

C so 26,000 years is half-lives.
The fraction that remains after half-lives is
That is 6.25% .
No formula used. We do not need for that.
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