# Radioactive substances decay exponentially. For example, a sample of Carbon-14(^{14}C)will lose half of its mass every 5730 years. (In other words, th

Radioactive substances decay exponentially. For example, a sample of Carbon$-14{\left(}^{14}C\right)$will lose half of its mass every 5730 years. (In other words, the half-life of ${14}_{C}$ is 5730 years.) Let A be the initial mass of the sample. Model the decay of ${}^{14}C$ using a discrete-time model... (a) using $\delta t=5730$ years. (b) using Δt=1year.

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aprovard

Given:
A initial mass
Half-life of ${}^{\left\{4\right\}}C$ is 5730 years
(a)
$\delta t=5730$ years
Let m(t) represent the mass of the Carbon-14 after ¢ periods of 5730 years
Initinlly, the mass is equal to A.
(0) =A
After 1 period of 5730 years, the mass m(t-1) of 5730 years ago is divided by half as the halflife of Carbon-14 is 5730 years.
$m\left(t\right)=\frac{m\left(t-1\right)}{2}$ for t>0
Combining these two expressions, we than obtain:
(b)
$\delta t=1$ year
Let m(t) represent the mass of the Carbon-14 after t years.
Initinlly, the mass is equal to A.
m(0) =A
After 5730 years, the mass A is divided by half as the halfife of Carbou-14 is 5730 years.
$m\left(5730\right)=\frac{A}{2}$
Combining these two expressions, we then obtain:
 Or you could also use the formula $m\left(t\right)=A{\left(\frac{1}{2}\right)}^{\frac{t}{5730}}$ instead (formula for the halfiine), but the book claims that we need to use recurrence relations in this case (which makes is very hard to properly defined it in this case).