Problem 75 - Algebra and Trigonometry

Question

Radioactive Decay \(\quad\) A 15 -g sample of radioactive iodine decays in such a way that the mass remaining after \(t\) days is given by \(m(t)=15 e^{-0.087 t}\) where \(m(t)\) is measured in grams. After how many days is there only 5 g remaining?

Step-by-Step Solution

Answer

It takes approximately 13 days for the mass to decrease to 5 g.

Step 1: Identify the given function and parameters

We are given the function for the radioactive decay of iodine as \( m(t) = 15e^{-0.087t} \), where \( m(t) \) represents the mass remaining after \( t \) days. The problem asks us to find the value of \( t \) when there is only 5 g remaining.

Step 2: Set up the equation

We need to find the time \( t \) when the mass \( m(t) \) is 5 g. Start by setting \( m(t) \) to 5 and plug it into the equation: \( 5 = 15e^{-0.087t} \).

Step 3: Isolate the exponential term

Divide both sides of the equation by 15 to isolate the exponential term: \( \frac{5}{15} = e^{-0.087t} \). Simplifying gives \( \frac{1}{3} = e^{-0.087t} \).

Step 4: Apply natural logarithm

To solve for \( t \), take the natural logarithm of both sides of the equation: \( \ln\left(\frac{1}{3}\right) = \ln(e^{-0.087t}) \). This can be simplified to \( \ln\left(\frac{1}{3}\right) = -0.087t \).

Step 5: Solve for t

Divide both sides of the equation by \(-0.087\) to solve for \( t \): \( t = \frac{\ln\left(\frac{1}{3}\right)}{-0.087} \). Calculate this result to find \( t \approx 12.6 \).

Step 6: Interpret the result

The calculated \( t \approx 12.6 \) represents the number of days it takes for the mass to decrease to 5 g. Since time is generally measured in whole days, you might consider \( t \approx 13 \) days.

Key Concepts

Understanding Exponential Decay

Exponential decay is a process where a quantity decreases at a rate proportional to its current value. In simpler terms, the larger the quantity is, the faster it decreases. This behavior is common in real-world phenomena like radioactive decay, where substances lose their mass over time.
In the given problem, the mass of iodine decreases exponentially, as shown by the formula:

• \( m(t) = 15 e^{-0.087t} \)

This equation tells us how much iodine remains after \( t \) days. The negative exponent signifies the decay.
Over time, the mass reduces, showcasing the essence of exponential decay. The initial mass is multiplied by an exponential function that continuously reduces the total mass.

Natural Logarithm and Its Application

The natural logarithm, denoted as \( \ln \), is the inverse operation of the exponential function with base \( e \). It's essential when solving problems involving growth and decay rates, like radioactive decay.
In our problem, the step is:

• Take the natural logarithm of both sides of the equation: \( \ln\left(\frac{1}{3}\right) = \ln\left(e^{-0.087t}\right) \)

This helps us simplify the process by bringing down the exponent \( -0.087t \) in front due to the logarithm rules. Thus, we get:

• \( \ln\left(\frac{1}{3}\right) = -0.087t \)

By rearranging this equation, we readily solve for \( t \). It's a crucial step to determine the timeframe of the decay.

Role of Exponential Function in Decay

Exponential functions are vital in describing processes that change at rates proportional to their value. An exponential function looks like \( e^{x} \), where \( e \) is a mathematical constant approximately equal to 2.71828. In decay processes, it takes the form of \( e^{-x} \), indicating a reduction.
For the iodine decay problem, the function is:

• \( m(t) = 15e^{-0.087t} \)

Here, the exponential function \( e^{-0.087t} \) regulates the decay rate. As time (\( t \)) increases, the term \( e^{-0.087t} \) becomes smaller, leading to a reduction in mass. This function effectively models the natural decay of substances over time.

Solving the Half-Life Problem

A half-life problem involves finding the time it takes for a quantity to reduce to half its initial value. Though our specific problem asked for the time to drop to 5 grams, it utilizes similar principles used in calculating half-life scenarios.
To address such problems, we:

• Identify the exponential decay function
• Determine the remaining mass
• Solve for time using logarithms

This procedure is what we did to find that approximately 12.6 days are required for the iodine to decay from 15 grams to 5 grams. Understanding half-life is crucial in sciences dealing with decay, such as chemistry and physics, allowing prediction of material behavior over time.