# Math Help - Carbon-14 Half-life word problem

1. ## Carbon-14 Half-life word problem

Anyone know how to go about solving a problem like this?

Yesterday the Holy Grail was found in my lawn. Knowing the Grail must be 2,000 yrs old and Carbon-14's 1/2 Life is 5,700 yrs, what % must be gone to be real

2. Originally Posted by cjmac87
Anyone know how to go about solving a problem like this?

Yesterday the Holy Grail was found in my lawn. Knowing the Grail must be 2,000 yrs old and Carbon-14's 1/2 Life is 5,700 yrs, what % must be gone to be real
What a coincidence! i found the Grail in my backyard just last week. mine is the real one of course.

Let $A(t)$ be the amount of carbon-14 remaining at time t, let $A_0(t)$ be the original amount of carbon-14. since we lose carbon-14 in an exponential decay, we have that:

$A = A_0e^{-rt}$

where $r$ is the rate of decay and $t$ is the time elapsed.

now, we know that $r = \frac {\ln 2}{t_h}$ where $t_h$ is the half-life

thus we have:

$A = A_0e^{\frac {\ln 2}{5700}t}$

now assume we have 1 unit to begin with. then the A remaining after 2000 years (in the decimal form of a percentage) is given by:

$A = e^{\frac {\ln 2}{5700}(2000)}$

now calculate that

3. Originally Posted by cjmac87
Anyone know how to go about solving a problem like this?

Yesterday the Holy Grail was found in my lawn. Knowing the Grail must be 2,000 yrs old and Carbon-14's 1/2 Life is 5,700 yrs, what % must be gone to be real
The amount remaining given an initial amount $A_0$ after time $t$ if $t_{HL}$ is the
half life is:

$
A(t)=A_0 2^{-t/t_{HL}}
$

In this case let $A_0=100$.

RonL