Problem 7 - Calculus with Concepts in Calculus

Question

Suppose the amount \(f(t)\) of a radioactive substance decays exponentially, so that \(f(t)=f(0) e^{k t}\) for some negative constant \(k\). a. Show that the amount decreases by half in any time interval of duration \(-(\ln 2) / k .\) Thus \(-(\ln 2) / k\) is the half-life \(h\) b. Show that \(f(t)=f(0)\left(\frac{1}{2}\right)^{t / / h}\).

Step-by-Step Solution

Answer

The amount decreases by half in time \(h = -\frac{\ln 2}{k}\), and \(f(t) = f(0) \left(\frac{1}{2}\right)^{t/h}\).

Step 1: Understand the Exponential Decay Formula

Given that the radioactive substance decays exponentially, the amount at time \(t\) is expressed as \(f(t) = f(0) e^{kt}\), where \(f(0)\) is the initial amount and \(k\) is a negative constant.

Step 2: Set Up the Equation for Half-Life

We want to show that the amount decreases by half over a specific time interval, called the half-life \(h\). Assume that at time \(h\), the amount is half of what it was initially: \(f(h) = \frac{1}{2}f(0)\). Substituting into the decay formula gives \(\frac{1}{2}f(0) = f(0) e^{kh}\).

Step 3: Solve the Half-Life Equation

Divide both sides by \(f(0)\) to get \(\frac{1}{2} = e^{kh}\). Take the natural logarithm of both sides to solve for \(kh\): \(\ln\left(\frac{1}{2}\right) = kh\). This simplifies to \(-\ln 2 = kh\), so \(h = \frac{-\ln 2}{k}\), showing the expression for half-life.

Step 4: Express \(f(t)\) in Terms of Half-Life

Next, express \(f(t)\) using the half-life \(h\). We know that \(h = \frac{-\ln 2}{k}\). Substitute this into the original decay equation \(f(t) = f(0) e^{kt}\).

Step 5: Substitute \(h\) into the Decay Formula

Replace \(k\) in \(kt\) with \(-\frac{\ln 2}{h}\), obtaining \(f(t) = f(0) e^{\left(-\frac{\ln 2}{h}\right)t}\). This can be rewritten using the properties of exponents: \(f(t) = f(0) \left(e^{-\ln 2}\right)^{t/h}\).

Step 6: Simplify the Expression

Since \(e^{-\ln 2} = \left(\frac{1}{2}\right)\), substitute back to get \(f(t) = f(0)\left(\frac{1}{2}\right)^{t/h}\). This expression shows the exponential decay of \(f(t)\) related to the half-life \(h\).

Key Concepts

Exponential Decay

Exponential decay describes a process where the quantity of something decreases at a rate proportional to its current value. This is commonly seen with radioactive materials, where the amount of substance diminishes exponentially over time. In the context of our problem, we use the mathematical formula \(f(t) = f(0) e^{kt}\), where \(f(0)\) represents the initial amount of the substance, \(t\) is time, and \(k\) is a negative constant that governs the rate of decay.
Understanding exponential decay is crucial because it models various natural and artificial processes. For instance:

• Substances reducing in mass over time (like a melting ice cube).
• The cooling of a heated object placed in a naturally colder environment.
• The depreciation of an asset over years.

The key characteristic of exponential decay is its constant proportionality, meaning the amount reduced in any given period is consistent relative to the amount present at the start of that period.

Radioactive Decay

Radioactive decay is a type of exponential decay that applies specifically to unstable atomic nuclei. Over time, these nuclei release energy and particles, transforming into a more stable state. This transformation occurs at a predictable rate, defined by the substance's half-life.
The half-life is the period required for half of the radioactive material to disintegrate. In the exponential decay equation \(f(t) = f(0) e^{kt}\), this is represented by the time interval \(-\frac{\ln 2}{k}\), where \(k\) is the decay constant.
In practical terms, understanding radioactive decay allows us to:

• Dater ancient artifacts through carbon-14 decay.
• Manage nuclear waste by knowing how long it will remain hazardous.
• Develop medical treatments and imaging techniques using radioactive isotopes.

The half-life concept is especially useful because it allows predictions about how long a particular sample will remain active, or how quickly it will diminish to a safe level.

Logarithmic Functions

Logarithmic functions are the inverse of exponential functions, and they play a vital role in solving exponential decay problems. When dealing with radioactive decay, we often need to convert the exponential form into a more workable form. This is where the natural logarithm (\(\ln\)) comes in handy. Consider when we want to find the half-life of a radioactive substance. Starting from the equation \(\frac{1}{2} = e^{kh}\), we take the natural logarithm of both sides, resulting in \(\ln\left(\frac{1}{2}\right) = kh\), simplifying to \(-\ln 2 = kh\). The natural logarithm helps us solve for \(h\) by isolating it on one side of the equation:

• Break down exponential terms for easier manipulation.
• Provide a linear form to solve for variables.
• Translate multiplicative relationships into additive ones, simplifying complex problems.

Understanding logarithms is essential not only in radioactive decay but in various scientific and financial applications where growth or decay processes need to be understood and predicted.