Math Snap

STEP 1

What is this asking?
We're figuring out how old a fossil is based on how much carbon-14 is left, knowing that carbon-14 decays over time.
Watch out!
Don't mix up the initial amount with the current amount.
Also, make sure to round to the nearest year at the very end!

STEP 2

1. Set up the equation
2. Isolate the exponential
3. Solve for $t$

STEP 3

The problem says the fossil has \textbf{28%} of the carbon-14 of a living sample.
This means $A(t)$ is $0.28$ times the initial amount, $A_0$.
So we can write $A(t) = 0.28 \cdot A_0$.

STEP 4

We know the formula for carbon-14 decay is $A(t) = A_0 e^{-0.00012t}$.
Let's substitute the expression for $A(t)$ we just found:
$$0.28 \cdot A_0 = A_0 e^{-0.00012t}$$

STEP 5

To get the exponential term by itself, we can divide both sides of the equation by $A_0$.
This is valid since $A_0$ represents the initial amount of carbon-14, which is a positive value.
$$\frac{0.28 \cdot A_0}{A_0} = \frac{A_0 e^{-0.00012t}}{A_0}$$ $$0.28 = e^{-0.00012t}$$

STEP 6

To get $t$ out of the exponent, we can take the natural logarithm (ln) of both sides:
$$\ln(0.28) = \ln(e^{-0.00012t})$$

STEP 7

Remember that $\ln(e^x) = x$, so we have:
$$\ln(0.28) = -0.00012t$$

STEP 8

Now, we can divide both sides by $-0.00012$ to solve for $t$:
$$t = \frac{\ln(0.28)}{-0.00012}$$

STEP 9

Using a calculator, we find:
$$t \approx \frac{-1.272965675}{-0.00012}$$ $$t \approx 10608.04729$$

STEP 10

The problem asks us to round to the nearest year, so our final answer is approximately $\textbf{10608}$ years.

The age of the sample is approximately 10608 years.