Chapter 8.8, Problem 14HP

To determine

To find: The decay equation.

Answer to Problem 14HP

The decay equation is $y=a{e}^{-1.2\times {10}^{-4}t}$ .

Explanation of Solution

Given information:

Calculation:

Consider the case of carbon $14$ that is present in living and dead organisms.

Carbon $14$ decays exponentially at a fixed rate. This is used to determine the life of fossil.

The half-life of carbon $14$ is $5.730$ years.

The decay of carbon $-14$ can be modeled by the following formula is below,

  $y=e^{-kt}$

Where, $y$ is the amount of substance after decay and $a$ is the initial amount of radioactive.

Substance, $k$ is the decay constant and $t$ is the time of decay.

Therefore, the constant is

  $0.5a=a{e}^{-k=5730}$

  $\begin{array}{l}{e}^{-k\times 5730}=0.5\\ -k\times 5730=In(0.5)\\ k=\frac{-0.693}{-5730}\\ =1.2\times {10}^{-4}\end{array}$

Therefore, the decay equation is $y=a{e}^{-1.2\times {10}^{-4}t}$ .