Problem 99 - Precalculus

Question

Carbon dating is a method used to determine the age of a fossil or other organic remains. The age \(t\) in years is related to the mass \(C\) (in milligrams) of carbon 14 through a logarithmic equation: $$t=-\frac{\ln \left(\frac{C}{500}\right)}{0.0001216}$$ How old is a fossil that contains 100 milligrams of carbon \(14 ?\)

Step-by-Step Solution

Answer

The fossil is approximately 13,234 years old.

Step 1: Understand the Given Values

We are given that the fossil currently has 100 milligrams of carbon-14. In the formula, this means we substitute \( C = 100 \). The initial amount (before decay) of carbon-14 is 500 mg.

Step 2: Plug the Values into the Formula

Substitute \( C = 100 \) into the given formula: \[ t = -\frac{\ln \left(\frac{100}{500}\right)}{0.0001216} \]

Step 3: Simplify the Fraction Inside the Logarithm

Calculate the fraction inside the logarithm:\[ \frac{100}{500} = 0.2 \]

Step 4: Calculate the Natural Logarithm

Find the natural logarithm of the calculated fraction:\[ \ln(0.2) \approx -1.6094 \]

Step 5: Substitute and Simplify

Plug \( \ln(0.2) \approx -1.6094 \) back into the equation:\[ t = -\frac{-1.6094}{0.0001216} \]Simplify to find \( t \).

Step 6: Compute the Result

Calculate the final operation:\[ t \approx \frac{1.6094}{0.0001216} \approx 13234 \]Thus, the fossil is approximately 13,234 years old.

Key Concepts

Logarithmic Equations

Logarithmic equations like the one used in carbon dating express relationships involving logarithms. These equations are particularly useful when dealing with exponential decay problems, such as the decay of carbon-14 in fossils. The logarithm essentially helps us solve for unknowns, in this case, time "t." In the equation \[-\frac{\ln \left(\frac{C}{500}\right)}{0.0001216}\], the logarithmic portion, \(\ln \left(\frac{C}{500}\right)\), is central.
This part of the equation is derived from the general property of logarithms, which states that \(\ln(a/b) = \ln a - \ln b\). Understanding this property allows us to make sense of how the carbon mass ratio transforms logarithmically to solve for the fossil's age.

• Logarithms help us convert multiplication or division problems into addition or subtraction ones, which are easier to solve.
• In carbon dating, exponential decay is modeled using logarithms because the rate of decay reduces the carbon mass exponentially over time.

Natural Logarithm

The natural logarithm, denoted as \(\ln\), is a special logarithm with the base "e," a mathematical constant approximately equal to 2.71828. It often comes up in calculus and exponential growth or decay contexts, like carbon dating. The formula for carbon dating involves \(\ln\), indicating its reliance on exponential functions.
For instance, \(\ln(0.2)\) was calculated to determine the logarithmic relationship between the remaining carbon and its original amount.

• The natural logarithm is used in situations where things change in relation to their current state, such as radioactive decay.
• It is preferred in scientific applications because of its mathematical properties involving rate of change, fitting the pattern of natural phenomena.
• In carbon dating, using \(\ln\) makes it possible to work directly with the proportions of carbon remaining, simplifying the computation to find time.

Radioactive Decay

Radioactive decay refers to the process by which unstable atomic nuclei lose energy by emitting radiation. Carbon-14, a radioactive isotope, spontaneously decays over time, which is crucial for understanding its use in aging fossils. As it decays, the quantity of carbon-14 decreases exponentially, leading to the formation of nitrogen-14.
This natural process is the basis of carbon dating, where the exponential decay equation relates the original and remaining amounts of carbon-14 to time, allowing us to accurately estimate the age of organic materials.

• Radioactive decay follows a predictable pattern, described mathematically as an exponential decay, illustrated in carbon dating equations.
• This decay provides a natural "clock," making it possible to measure how long it has been since the organism was last alive.
• Understanding decay rates is key in many scientific fields, not just in dating ancient artifacts.