Problem 87 - Algebra and Trigonometry

Question

Carbon Dating The age of an ancient artifact can be determined by the amount of radioactive carbon-14 remaining in it. If \(D_{0}\) is the original amount of carbon- 14 and \(D\) is the amount remaining, then the artifact's age \(A\) (in years) is given by $$ A=-8267 \ln \left(\frac{D}{D_{0}}\right) $$ Find the age of an object if the amount \(D\) of carbon \(-14\) that remains in the object is 73\(\%\) of the original amount \(D_{0}\) .

Step-by-Step Solution

Answer

The artifact is approximately 2588 years old.

Step 1: Understanding the Problem

We need to determine the age of an artifact using the formula for carbon dating. The formula relates the original amount of carbon-14 ( D_{0} ) to the remaining amount ( D ) and gives the age A in years. We are given that D is 73% of D_{0} .

Step 2: Set Up the Equation

We'll use the given formula to find the age:\[ A = -8267 \ln \left(\frac{D}{D_{0}}\right) \]Given that D = 0.73D_{0}, substitute D/D_{0} with 0.73 in the equation.

Step 3: Substitution in the Formula

Substitute the value 0.73 for D/D_{0} into the formula:\[ A = -8267 \ln(0.73) \]

Step 4: Calculation of Natural Logarithm

Calculate the natural logarithm of 0.73 using a calculator:\[ \ln(0.73) \approx -0.313 \]

Step 5: Complete the Calculation

Now, multiply both values to find the age:\[ A = -8267 \times (-0.313) \approx 2587.671 \]

Step 6: Round the Answer

Round the result to a reasonable degree of accuracy, typically the nearest whole number. Here, round 2587.671 to 2588 .

Key Concepts

Radioactive Decay

Radioactive decay is a natural process where an unstable atomic nucleus loses energy by emitting radiation. In carbon dating, this process is crucial because it allows us to determine the age of ancient artifacts. When a living organism dies, it stops absorbing carbon-14, a radioactive isotope of carbon. Over time, the carbon-14 decays, changing into a more stable form of carbon. By measuring the amount of carbon-14 left in an object, scientists can calculate how long it has been since the organism died. This is because the rate at which carbon-14 decays is consistent and well understood.

Carbon-14

Carbon-14 is a radioactive isotope of carbon with an atomic nucleus containing 6 protons and 8 neutrons. It is naturally found in the atmosphere and absorbed by living organisms. After an organism dies, it no longer absorbs carbon-14, and the existing carbon-14 atoms start to decay. This decay provides a "clock" that can be used to calculate the time elapsed since the organism's death, a process known as radiocarbon dating. Carbon-14 has a half-life of about 5,730 years, meaning half of the original carbon-14 will have decayed in this period. This characteristic makes it invaluable for dating archaeological finds.

Natural Logarithm

The natural logarithm, denoted as \(\ln\), is a special logarithm with the base \(e\), where \(e\) is an irrational number approximately equal to 2.71828. It is used to solve problems involving exponential growth or decay, like carbon dating. By taking the natural logarithm of the ratio between the remaining and the original amount of carbon-14, we can easily find the age of an artifact. The formula used in carbon dating \(A = -8267 \ln(\frac{D}{D_0})\) involves a natural logarithm to help model the exponential decay of carbon-14 over time.

Exponential Functions

Exponential functions describe situations where growth or decay rates are proportional to the current quantity. In carbon dating, the decay of carbon-14 is modeled by an exponential function. This form is chosen because the rate at which \(\text{carbon-14}\) decays is proportional to its current amount. The exponential function \(D = D_0 e^{kt}\) can characterize this process, where \(k\) is the decay constant and \(t\) is time. However, for practical dating, the natural logarithm version \(A = -8267 \ln(\frac{D}{D_0})\) is used, making calculations manageable and straightforward.