Problem 23 - Calculus Concepts

Question

Carbon-14 Decay The rate at which 100 grams of \({ }^{14} \mathrm{C}\) is decaying can be modeled as \(r(t)=-0.027205\left(0.998188^{7}\right)\) grams per year where \(t\) is the number of years since the 100 grams began decaying. a. How much of the \({ }^{14} \mathrm{C}\) will decay during the first 1000 years? During the fourth 1000 years? b. How much of the \({ }^{14} \mathrm{C}\) will eventually decay?

Step-by-Step Solution

Answer

27.205 grams decay in both the first and fourth 1000 years, and eventually all 100 grams will decay.

Step 1: Understanding the Rate Function

The given rate function for the decay of Carbon-14 is \( r(t) = -0.027205 \times (0.998188)^7 \). This indicates the amount of \( ^{14}C \) that decays per year as a function of time \( t \). The negative sign represents that the mass is decreasing over time.

Step 2: Finding Amount Decayed Over 1000 Years

To find the total decay during the first 1000 years, integrate the rate function \( r(t) \) from \( t = 0 \) to \( t = 1000 \): \[ \int_0^{1000} r(t) \, dt. \] Substitute \( r(t) = -0.027205 \times (0.998188)^7 \) into the integral and solve. The constant factor can come outside the integral, giving \( -0.027205 \times 1000 \times (0.998188)^7 \). Calculate this value.

Step 3: Calculating First 1000 Years

Calculate \( -0.027205 \times 1000 \times (0.998188)^7 \) which results in approximately \( -27.205 \) grams decayed in the first 1000 years.

Step 4: Finding Amount Decayed During the Fourth 1000 Years

Now, find the decay from year 3000 to year 4000 using the same integration principle: \[ \int_{3000}^{4000} r(t) \, dt. \] This will also result in the same calculation: \( -0.027205 \times 1000 \times (0.998188)^7 \), approximately \( -27.205 \) grams decayed during the fourth 1000 years.

Step 5: Total Amount That Will Eventually Decay

To find the total amount that will eventually decay, consider that the decay continues indefinitely. This means, theoretically \( r(t) \) should be integrated from 0 to infinity. However, as \( t \to \infty \), the remaining mass approaches zero. Therefore, practically, the entire 100 grams will eventually decay, leaving zero mass.

Key Concepts

Rate of Decay

The rate of decay refers to the speed at which a radioactive substance, such as Carbon-14, loses its mass over time. It's a dynamic process described mathematically by a rate function, which in this exercise is defined as \[ r(t) = -0.027205 \times (0.998188)^7 \]The negative sign in the rate function indicates a loss of material. Over time, the rate reflects the mass of Carbon-14 that reduces annually. Think of it as a countdown timer for decay, signalling how fast the substance is diminishing. In this context, understanding the rate of decay is essential because it helps us calculate how much of Carbon-14 remains over a given period.
This rate is constant, meaning that each year the substance decays at the same rate. Here, "rate" helps us model the exact amount of Carbon-14 decaying year-by-year or over any specific time frame.

Integration

Integration is a fundamental concept in calculus. It allows us to calculate the total amount of a substance that decayed over a specific timeframe. In this exercise, to find the amount of Carbon-14 that decays during the first 1000 years, we integrated the rate function from \[ t = 0 \text{ to } t = 1000 \]. The integration of the decay rate \[ (t) = -0.027205 \times (0.998188)^7 \]from 0 to 1000 calculates the total loss in mass over those years. It's like adding up an infinite number of tiny pieces of decay over time to find the big picture.
For solving the problem, the integration process simplified to multiplying the decay rate by the duration, yielding an approximate calculation of \[ -27.205 \text{ grams} \].
In a similar manner, integrating from year 3000 to year 4000 gives us the decayed mass for that particular period. It's this step that transforms our rate function into a total value of decayed Carbon-14.

Exponential Decay

Exponential decay describes the manner in which the quantity of Carbon-14 decreases over time. In exponential decay, the rate at which the material decreases is proportional to its remaining quantity. This type of decay mathematically manifests itself with an equation where the decay rate is multiplied by a base raised to the power of time, i.e., \[ (0.998188)^t \].
Exponential decay is common in radioactive materials because they lose a fixed percentage of their quantity per time interval. It’s a graceful and predictable decline, not a linear or abrupt one. This exercise introduces the exponential decay process of \(^{14}C\)by demonstrating its consistent rate of loss. Thus, for any time frame, \((0.998188)^7\)signifies a decrease reflecting natural decay characteristics.

Mathematical Modeling

Mathematical modeling turns real-world decay phenomena into solvable equations. This exercise models the decay of Carbon-14 with a function representing its changing mass. Using mathematical expressions, we translate the physical process of atomic decay into something we can calculate and predict.
Through a rate function and integration, we gain insights into the future and past state of the material undergoing decay. It offers a systematic approach to computing the decay at any point. This modular representation simplifies complexity, converting natural decay patterns into understandable mathematical terms.
Mathematical models like in this exercise use common, logical patterns. They help to hypothesize patterns of decay over indefinite time spans, and ensure we calculate the remaining and decayed quantity effectively.