Problem 58 - Calculus with Concepts in Calculus

Question

Suppose that a living organism died \(t\) years ago. The number \(t\) frequently can be assessed by carbon-14 dating. If \(p\) percent of the original \(\mathrm{C}^{14}\) in the organism is now present, then \(t\) is given approximately by $$ t=-\frac{\ln (p / 100)}{0.000124} $$ Find the approximate age of each of the following objects with the given value of \(p\). a. mammal tusk, where \(p=1\) b. wooden post, where \(p=60\)

Step-by-Step Solution

Answer

The mammal tusk is approximately 37,138 years old, and the wooden post is approximately 4,120 years old.

Step 1: Insert Given Values for the Mammal Tusk

For the mammal tusk, substitute the given value of \( p = 1 \) into the formula: \[ t = -\frac{\ln(1 / 100)}{0.000124} \]

Step 2: Calculate the Natural Logarithm for the Mammal Tusk

Calculate \( \ln(1/100) \). Since \( 1/100 = 0.01 \), we need to find: \[ \ln(0.01) \] Using a calculator, \( \ln(0.01) \approx -4.60517 \).

Step 3: Solve for the Age of the Mammal Tusk

Now substitute \( \ln(0.01) = -4.60517 \) back into the equation: \[ t = -\frac{-4.60517}{0.000124} \] Simplifying, we get:\[ t \approx \frac{4.60517}{0.000124} \approx 37138.47 \] So, the mammal tusk is approximately 37,138 years old.

Step 4: Insert Given Values for the Wooden Post

For the wooden post, substitute the given value of \( p = 60 \) into the formula: \[ t = -\frac{\ln(60 / 100)}{0.000124} \]

Step 5: Calculate the Natural Logarithm for the Wooden Post

Calculate \( \ln(60/100) \). Since \( 60/100 = 0.6 \), we need to find: \[ \ln(0.6) \] Using a calculator, \( \ln(0.6) \approx -0.5108256 \).

Step 6: Solve for the Age of the Wooden Post

Substitute \( \ln(0.6) = -0.5108256 \) back into the equation: \[ t = -\frac{-0.5108256}{0.000124} \] Simplifying, we get:\[ t \approx \frac{0.5108256}{0.000124} \approx 4119.56 \] So, the wooden post is approximately 4,120 years old.

Key Concepts

Understanding the Natural Logarithm

The natural logarithm, represented as \( \ln \), is a logarithm to the base \( e \), where \( e \) is approximately 2.71828. Unlike regular logarithms that might use bases such as 10 or 2, the natural logarithm uses \( e \), a constant that emerges often in mathematics and nature. It's a crucial part of many calculus and exponential growth calculations.
When we say \( \ln(x) \), it is the power to which \( e \) must be raised to get \( x \). For example, if \( \ln(2) \approx 0.693 \), it means \( e^{0.693} \approx 2 \).\( \ln \) is essential in decay problems, like the carbon-14 dating formula, because it transforms multiplicative scales into additive ones. This makes complex exponential decay rates, like those of carbon-14 in an organism, easier to solve.

Exploring the Decay Formula

The decay formula used in carbon-14 dating provides us with a means to estimate the age of archeological artifacts by determining how much carbon-14 remains. The specific formula, \[ t = -\frac{\ln(p/100)}{k} \] where \( t \) is the time in years, \( p \) is the percentage of carbon-14 remaining, and \( k \) is the decay constant (0.000124 in this case). This formula essentially calculates the amount of time needed for the remaining carbon-14 to have decayed from its original quantity to \( p \) percent.
The decay formula leverages the natural logarithm, which helps deal with the exponential nature of radioactive decay efficiently. Because decay processes don't follow a simple linear pattern but rather an exponential one, the formula's use of the natural logarithm is key in converting this curved decay into a solvable linear problem.

Estimating the Age of Artifacts

To estimate how old an object is using carbon-14 dating, we follow a consistent process. First, measure the amount of carbon-14 remaining in the specimen and express this as a percentage of the original. Then, substitute this percentage into the decay formula given as:\[ t = -\frac{\ln(p/100)}{0.000124} \]
For example, if we have a mammal tusk with \( p = 1 \), this indicates that only 1% of the original carbon-14 is left. Plugging \( p \) into the formula allows us to find \( t \), giving us an estimated age of 37,138 years.
Similarly, for a wooden post with \( p = 60 \), we substitute 60 into the formula. This contributes to an estimated age of approximately 4,120 years. By using these calculations, archaeologists and scientists can determine how long ago something died with remarkable precision, opening a window into the past.