Does anyone know how to solve this question?

Question

An isotope of a radioactive element has half-life equal to 7 thousand years.

Imagine a sample that is so old that most of its radioactive atoms have decayed, leaving just 4 percent of the initial quantity of the isotope remaining.

How old is the sample?

Give your answer in thousands of years, correct to one decimal place.

Answer 1

Let I = initial amount
A = current amount at time y

A = I (1/2)^(y/7)
A = 0.04I

0.04I = I (1/2)^(y/7)

0.04 = (1/2)^(y/7)

ln (0.04) = (y/7) ln(1/2)
y = 7ln(0.04)/ln(0.5)
y = 32.5 thousand years

Answer 2

First, some editorial comment. Half-life is a crutch that chemists use because they were too dumb to learn pre-calculus. It should be replaced by a decay constant. And you can tell your chemistry teacher that.

In each half-life, 50 percent of the matter at the start of the half-life interval will disappear by the end of the half-life interval. So if we started with 100 units of matter, at the end of 1 half-life, we have 50, 2 half-lives, 25 and so-on. To find the number of half lives we have, we can solve the equation 25 = 2^x. The 25 represents the inverse ratio of initial to final mass of atoms or 100/4. Solving this, we have x of about 4.7. So the age is 7000 years x 4.7, or 33,000 years.

Answer 3

half-life = 7000 = 50%
100% = 14000
if 4% reminds from the isotope's life, it lived 96% from 14000
= 13.440 years.

In thousands of years, correct to one decimal place = 13.4

Answer 4

Decay rate of 1/2 each 7000 years.
When does x(1/2)^t = .04*x?
(1/2)^t=.04 Log of both sides:
t*log1/2=log.04
t=(log.04)/(log .5)
t is how many times the substance has halved, so because it halves every 7000 years, 7000*t is your final answer.

Answer 5

N = Nz*e^(-kt)
To find k, Set N/Nz to 1/2 and t to 7000
-ln(1/2)/7000 = k (99.02e-6)
then N/Nz = 0.04 now solve for t
-ln(0.04)/99.02e-6 = t (32507 years)
or 32.5 thousand years

Answer 6

remaining life L = 2^(-k/7), k in thousand years

0.04 = 2^(-k/7)

log(0.04) = -k/7*log(2)

k = -7*log(.04)/log(2) = 32.5 thousand years

Answer 7

solve:
(0.5^x) = .04
x = number of half lives

then multiply x by 7000 years to get the total years needed

Answer 8

why would I want to deprive you of the brain exercise to come up with this simple solution and deprive you of the accomplishment you will feel??

Answer 9

55.20