In the formula A(t) = A0ekt, A is the amount of radioactive material remaining from an
initial amount A0 at a given time t, and k is a negative constant determined by the nature
of the material. An artifact is discovered at a certain site. If it has 72% of the carbon-14 it
originally contained, what is the approximate age of the artifact? (carbon-14 decays at the
rate of 0.0125% annually.) (Round to the nearest year.)
Ah, the age-old question of dating artifacts. Well, let's put on our clown hats and calculate!
First, we need to figure out the value of k, the negative constant. Since we know that carbon-14 decays at a rate of 0.0125% annually, we can convert that to a decimal by dividing it by 100.
0.0125% = 0.000125
Now, we have the formula A(t) = A0e^(kt). We're given that the artifact has 72% of the original carbon-14 it contained, so A(t) = 0.72A0.
Solving for the age, we get:
0.72A0 = A0e^(k*t)
Now, let's isolate that exponential term by dividing both sides by A0:
0.72 = e^(kt)
To get rid of the exponential term, we need to take the natural logarithm (ln) of both sides:
ln(0.72) = ln(e^(kt))
Now, since ln(e^(kt)) simplifies to kt, we can solve for t:
t = ln(0.72) / k
Now, substituting the value of k as 0.000125, we can calculate the approximate age.
t = ln(0.72) / 0.000125
Calculating this, we get t ≈ 4916.012 years.
So, based on my clown calculations, the approximate age of the artifact is about 4916 years. Keep in mind, this is rounded to the nearest year, so it's just an approximation.
To determine the approximate age of the artifact, we can use the radioactive decay formula A(t) = A0 * e^(kt).
In this case, we have 72% of the original amount of carbon-14 remaining, which means A(t) = 0.72A0.
We know that the decay rate is 0.0125% annually, which means k = -0.0125.
Now, we can substitute these values into the equation and solve for t:
0.72A0 = A0 * e^(-0.0125t)
Dividing both sides of the equation by A0:
0.72 = e^(-0.0125t)
To solve for t, we need to isolate the exponential term. Taking the natural logarithm (ln) of both sides of the equation will allow us to do that:
ln(0.72) = -0.0125t
Now, we can solve for t by dividing both sides of the equation by -0.0125:
t = ln(0.72) / -0.0125
Using a calculator, we find:
t ≈ 180.118
Rounding to the nearest year, the approximate age of the artifact is 180 years.
You are solving
.72 = 1(e^(-.0125t)
ln .72= ln (e^(-.0125t))
-.0125t = ln .72
t = ln.72/-.0125 = appr. 26.28 years
radioactive=decay
convert % to decimal.
72%=0.72, 0.0125%=0.000125
A=Aoe^-kt
0.72A=Aoe^-0.000125t
0.72=e^-0.000125t
ln 0.72=ln e^-0.000125t
ln 0.72=-0.000125t
ln 0.72/-0.000125=t
t=2628