Problem 15 - Algebra and Trigonometry

Question

An artifact originally had 16 grams of carbon- 14 present. The decay model \(A=16 e^{-0.000121 t}\) describes the amount of carbon- 14 present after \(t\) years. Use this model to solve Exercises \(15-16\) How many grams of carbon- 14 will be present in 5715 years?

Step-by-Step Solution

Answer

The amount of carbon-14 that will be present in 5715 years is the solution of \(A = 16 e^{-0.000121 * 5715}\).

Step 1: Understand the Decay Model

The decay model tells us about the amount of carbon-14 remaining after a period of time. In the given model, \(A=16 e^{-0.000121 t}\), where \(A\) is the amount of carbon-14 present after \(t\) years, \(16\) is the initial amount of carbon-14, the term \(e^{-0.000121 t}\) represents the exponential decay (with \(e\) being the base of natural logarithms), and the rate of decay is provided by the coefficient of \(t\) in the power of \(e\), which is \(-0.000121\).

Step 2: Substitution

Substitute the value of \(t\) as \(5715\) into the decay model to find out the amount of carbon-14 after \(5715\) years. So, \(A = 16 e^{-0.000121 * 5715}\).

Step 3: Perform the Calculation

Calculate the amount of carbon-14 remaining by solving the equation \(A = 16 e^{-0.000121 * 5715}\).

Step 4: Interpret the Result

The solution gives the amount of carbon-14 that will be present in 5715 years.

Key Concepts

Exponential Decay

Exponential decay is a process in which a quantity decreases at a rate proportional to its current value. In the context of carbon-14, this means that the amount of carbon-14 in a sample decreases by a fixed percentage each year. The decay model formula is generally given as A = A_0e^{-kt}, where:

• A is the amount of substance remaining after time t,
• A_0 is the initial quantity of the substance,
• k is the decay constant specific to the substance, and
• e is Euler's number, the base of natural logarithms.

The negative sign in the exponent highlights that the quantity is decreasing over time. To make this concept more tangible, let's look at an example. If you start with 16 grams of carbon-14, and it decays with a constant rate (given in the problem as 0.000121), in a given time frame, you would use the given model A=16e^{-0.000121t} to calculate how much carbon-14 remains after t years. For instance, after 5715 years, the amount of carbon-14 decreases exponentially based on this model.

Natural Logarithms

Natural logarithms are mathematical functions that are inverses of exponential functions with base e, where e approximately equals 2.71828. It is denoted as ln(x), with x being the argument of the function. Natural logarithms are particularly useful when dealing with exponential growth or decay since they can help isolate the exponent in exponential equations.
For example, if we have an equation of the form A = A_0e^{-kt}, taking the natural logarithm of both sides can help us solve for t if we know the values of A, A_0, and k. The equation would become ln(A) = ln(A_0) - kt, which can be rearranged to find the time t when a certain amount of a substance will remain. In essence, natural logarithms provide a transformative tool to work with and understand exponential relationships in various scientific fields, including radioactive decay and financial growth models.

Radioactive Dating

Radioactive dating is a technique used to date materials such as rocks or carbon, in which trace radioactive impurities were selectively incorporated when they formed. Carbon-14 dating, a form of radioactive dating, is used to estimate the age of carbon-bearing materials up to about 60,000 years old. The method is based on the principle of exponential decay of carbon-14, which has a known half-life of about 5730 years.
By measuring the amount of carbon-14 remaining in a sample and comparing it to the initial amount believed to be present at the time of death of the organism, scientists can calculate the time elapsed since the organism's death. The formula A=A_0e^{-0.000121t} is central to this process as it directly connects the remaining amount of carbon-14 (A) in the artifact with the time (t) since its last synthesis or death. Simplistically, if you found that an artifact contained half the amount of carbon-14 it started with, you could infer that it is approximately 5730 years old, the half-life of carbon-14. This is an invaluable tool in fields such as archaeology and geology for dating ancient organic remains.