# Exponential Decay

• Nov 17th 2009, 10:47 AM
littlesohi
Exponential Decay
How old is a piece of wood that has lost 90% of its carbon-14?

I know that there is only a presence of 10% but don't know how to find the decay rate (k).
• Nov 17th 2009, 11:45 AM
earboth
Quote:

Originally Posted by littlesohi
How old is a piece of wood that has lost 90% of its carbon-14?

I know that there is only a presence of 10% but don't know how to find the decay rate (k).

1. The half-life time of carbon-14 is 5730 years.

2. Use the equation:

$A(t)=A_0 \cdot e^{k \cdot t}$ to calculate k,

where A(t) is the present amount and $A_0$ the initial amount.

3. $\frac12 A_0 = A_0 \cdot e^{k \cdot 5730}~\implies~k= \dfrac{\ln\left(\frac12\right)}{5730}$

4. Solve the following equation for t:

$\frac1{10} A_0=A_0 \cdot e^{\frac{\ln\left(\frac12\right)}{5730} \cdot t}$

Spoiler:
I've got 19,035 years