Question

How to solve half life problem?

So I know that the problem is linear and not continuous. It has a decaying and has b=1-r . I have no idea how to start or solve the problem.

Is the equation `P=1+(5.471/1)^(1t)` ?

Answer 1

`12.32` years

Explanation:

Your equation is the one for compound interest rate on a principle amount. The math is similar (exponential function) but not the same arrangement. That is a good approach for this problem when given an annual rate.

The definition of a “half-life” is the time that it takes for `1/2` of the compound to decay. SO, we just need to find the point (number of years) at which a 5.471% decrease will result in 50% of the original material.

Relationship between concentration and time for a first-order reaction (radioactive decay) is:

`ln(([A]_t)/[A]_0)) = -kt`
where [A] is the concentration at time (t) and k is the rate constant.

Any quantity that grows or decays by a fixed percent at regular intervals is said to possess exponential growth or exponential decay.
there are two functions that can be easily used to illustrate the concepts of growth or decay in applied situations.

When a quantity grows by a fixed percent at regular intervals, the pattern can be represented by the functions:
Growth: `y = a(1+r)^x`
Decay: `y = a(1-r)^x`
WHERE:
a = initial amount before measuring growth/decay
r = growth/decay rate (often a percent)
x = number of time intervals that have passed
http://www.regentsprep.org/regents/math/algebra/ae7/expdecayl.htm

Applied to this problem we can write:
`0.5 = 1.0(1-0.05471)^(yr)`
`ln(0.5) = (yr) xx ln(0.94529)` ; `-0.693 = -0.05626 xx yr`

`yr = 12.32`

(12.26 is the actual reported tritium half-life)