The skeleton has lost 56 percent of their carbon-14. The constant proportionality is approximately k=0.000124. Estimate the age of the bones.
To estimate the age of the bones, we can use the formula for exponential decay:
N(t) = N₀ * e^(-kt),
where N(t) is the remaining amount of carbon-14 at time t, N₀ is the initial amount of carbon-14, k is the decay constant, and e is the base of the natural logarithm.
In this case, we are given that the skeleton has lost 56 percent of its carbon-14. This means that the remaining amount of carbon-14 is 100% - 56% = 44% = 0.44 of the initial amount (N₀).
We are also given that the constant proportionality (decay constant) is approximately k = 0.000124.
So, substituting these values into the decay equation, we have:
0.44N₀ = N₀ * e^(-0.000124t).
We can simplify this equation by canceling the N₀ on both sides:
0.44 = e^(-0.000124t).
To solve for t, we need to take the natural logarithm (ln) of both sides:
ln(0.44) = ln(e^(-0.000124t)).
Using the logarithmic property ln(a^b) = b * ln(a), the equation becomes:
ln(0.44) = -0.000124t * ln(e).
Since ln(e) equals 1, the equation further simplifies to:
ln(0.44) = -0.000124t.
Now, we isolate t by dividing both sides of the equation by -0.000124:
t = ln(0.44) / -0.000124.
Using a calculator, evaluate ln(0.44) / -0.000124 to find the estimated age of the bones.
(Note: The precise value will depend on the accuracy of the approximation and the rounding conventions used.)
if your constant is the k in
e^-kt, then you have
e^-0.000124t = 0.56
Now you can calculate the approximate age.
If by constant of proportionality you mean that that fraction is lost each year, then you have
0.999876^t = 0.56