SOLUTION: If 80​% of a radioactive element remains radioactive after 200 million​ years, then what percent remains radioactive after 700 million​ years? What is the half life of this e

Algebra.Com
Question 1202012: If 80​% of a radioactive element remains radioactive after 200 million​ years, then what percent remains radioactive after 700 million​ years? What is the half life of this element?
Found 4 solutions by ikleyn, Theo, greenestamps, josgarithmetic:
Answer by ikleyn(52830)   (Show Source): You can put this solution on YOUR website!
.
If 80​% of a radioactive element remains radioactive after 200 million​ years,
then what percent remains radioactive after 700 million​ years? What is the half life of this element?
~~~~~~~~~~~~~~~~~~~~~~~~ 

Let T be the half-life, in millions years.


We are given that

    0.8 = .


To find the half-life from here, take the logarithm base 10 of both sides.  You will get

    log(0.8) = ,     = 

    T =  = 621.2567  million years (the half-life).


The percent remaining after 700 millions years from the beginning is

     = 0.45795 = 45.795%  (rounded).

Solved.

------------

On radioactive decay,  see the lesson
    - Radioactive decay problems
in this site.

You will find many similar  (and different)  solved problems there.


/////////////////


The difference between my solution and the  @Theo' solution is that  @Theo
makes  TONS  of unnecessary calculations on the way,  while I make  NO  ONE  unnecessary calculation.

The method which I use (with the half-life decay formula) is ALWAYS preferable,
when half-life is given or half-life is under the question.

It is even not a subject to discuss - - - it is the way to follow.



Answer by Theo(13342)   (Show Source): You can put this solution on YOUR website!
formua you can use is f = p * (1+r) ^ n
f is the future value
p is the present value
r is the growth rate per time period.
2 + r is the growh factor per time period.
n is the number of time periods

the time periods are in millions of years.

after 200 million years, 80% of the element remains radioactive.
formula becomes:
.8 = 1 * (1 + r) ^ 200
divide both sides of the equation by 1 to get:
.8/1 = (1 + r) ^ 200
solve for (1 + r) to get:
(1 + r) = (.8/1) ^ (1 / 200) = .9988849044

that says that the growth factor is .9988849044 every million years.

to confirm, replace n in the original equation and solve for f to get:
f = 1 * .9988849044 ^ 200 = .8

now that you have the growth factor for every 1 million years, you can solve for the remaining percent after 700 million years.

the formula becomes f = 1 * .9988849044 ^ 700 = .4579467218 = 45/80%.

to find the half life, set f = .5 in the original equation and solve for n.

you will get:
.5 = 1 * .9988849044 ^ n
divide both sides of the eqution by 1 to get:
.5/1 = .9988849044 ^ n
simplify to get:
.5 = .9988849044 ^ n
take the log of both sides of the equation to get:
log(.5) = log(.9988849044 ^ n)
by log rule that says log(x^n) = n * log(x), this becomes:
log(.5) = n * log(.9988849044)
divide both sides of this equation by log(.9988849044) to get:
log(.5) / log(.9988849044) = n
solve for n to get:
n = 621.2567439 million years.

to confirm, replace n in the original eqution and solve for f to get:
f = 1 * .9988849044 ^ 621.2567439 = .5

the equation can be graphed as shown below:



Answer by greenestamps(13203)   (Show Source): You can put this solution on YOUR website!


For the first question, note that we do not have to determine the half life of the element.

We know that after 200 million years 80% or 0.8 of the original amount remains. 700 million years is 3.5 times as long, so the fraction remaining after 700 million years is (0.8)^(3.5) = 0.45795 to 5 decimal places.

1st ANSWER: Approximately 45.795% remains after 700 million years

(NOTE! Since radioactive decay is a statistical process and not a smooth mathematical process, keeping that many significant digits in the answer is probably unrealistic....)

For the second question, to find the half life, we can start by determining after how many half lives 80% of the original amount remains.






To several decimal places, that is 0.321928.

Then, since 80% remains after 200 million years, the half life in millions of years is

200/0.321928 = 621.257 to a few decimal places.
Answer by josgarithmetic(39621)   (Show Source): You can put this solution on YOUR website!
---------------------------------------------------------------------------------------------
If 80​% of a radioactive element remains radioactive after 200 million​ years, then ,...
---------------------------------------------------------------------------------------------

, if from model










If x is 700 million years, p is 1, then


46%


Half-Life
RELATED QUESTIONS

If 80% of a radioactive element remains radioactive after 200million years, then what... (answered by Edwin McCravy)
A certain radioactive element will decay at certain rate. If 50.0% of the original... (answered by josgarithmetic)
A radioactive substance is known to decay at a rate proportional to the amount present.... (answered by jorel1380)
A radioactive substance has a halflife of 120 years. If 200mg of substance was present,... (answered by ewatrrr)
A radioactive substance has a halflife of 120 years. If 200mg of substance was present,... (answered by josgarithmetic)
After 13 years, 2.1 pounds of radioactive material remains from a 7 pound sample. What is (answered by stanbon)
A radioactive substance decays exponentially. A scientist begins with 200 milligrams of a (answered by josgarithmetic)
A certain radioactive substance decays according to the formula m(t) =... (answered by oscargut)
A 200 g sample of radioactive polonium-210 has half-life of 138 days. a) Determine the (answered by josgarithmetic)