Problem 403 - Precalculus

Question

For the following exercises, use this scenario: A tumor is injected with 0.5 grams of lodine- \(125,\) which has a decay rate of 1.15\(\%\) per day. To the nearest day, how long will it take for half of the Iodine- 125 to decay?

Step-by-Step Solution

Answer

It takes about 60 days for half of the Iodine-125 to decay.

Step 1: Understand the Concept of Half-Life

Half-life is the time it takes for a substance to reduce to half its initial amount due to decay. In this problem, we're looking for the time it takes for half of the iodine to decay given a daily decay rate.

Step 2: Use the Exponential Decay Formula

The formula for exponential decay is \( N(t) = N_0 e^{-kt} \), where \( N(t) \) is the amount of substance left after time \( t \), \( N_0 \) is the initial amount, and \( k \) is the decay constant. First, calculate \( k \) using the given decay rate of 1.15\(\%\).

Step 3: Calculate the Decay Constant \( k \)

The decay rate as a daily percentage is 1.15\(\%\), equivalent to \( k = 0.0115 \) when expressed as a decimal.

Step 4: Set Up the Half-Life Equation

We need to find \( t \) such that \( N(t) = \frac{N_0}{2} \). Substituting into the decay formula: \( \frac{N_0}{2} = N_0 e^{-0.0115t} \).

Step 5: Solve for \( t \)

Divide both sides by \( N_0 \), which simplifies to \( \frac{1}{2} = e^{-0.0115t} \). Solve for \( t \) by taking the natural logarithm: \( \ln(\frac{1}{2}) = -0.0115t \). Therefore, \( t = \frac{\ln(\frac{1}{2})}{-0.0115} \).

Step 6: Calculate \( t \)

Plug into the logarithm function to determine \( t \): \( t = \frac{-0.693147}{-0.0115} \), yielding \( t \approx 60.27 \) days. Since time is usually rounded to the nearest whole number, in this context, it would be \( t \approx 60 \) days.

Key Concepts

Exponential Decay

Exponential decay is a process by which a quantity decreases at a rate proportional to its current value. This means that the more of the substance you have, the faster it decays. This concept is crucial in understanding how things diminish over time, particularly radioactive materials like iodine-125.
For exponential decay, we use the formula \( N(t) = N_0 e^{-kt} \). Here, \( N(t) \) represents the amount left at time \( t \), \( N_0 \) is the initial amount, and \( k \) is the decay constant.
Key points to remember:

• The rate of change is proportional to the current amount.
• Exponential decay results in a rapid decrease initially, which slows over time.

Understanding this process helps in calculating how long it will take a substance to reduce to a certain fraction of its original amount, such as half, which is the half-life.

Decay Constant

The decay constant \( k \) is a crucial part of the exponential decay formula. It represents the probability of decay per unit time. In simpler terms, it determines how quickly the substance decays.
For iodine-125 in the given scenario, the decay rate is given as 1.15\(\%\) per day. To convert this into a decay constant, we change the percentage into a decimal: \( k = 0.0115 \).
This simple conversion from percentage to decimal ensures that \( k \) can be used in calculations relating to time and decay, allowing us to find how long it takes for a substance to decay to half its original amount.
A higher \( k \) implies faster decay, while a lower \( k \) means slower decay.

Natural Logarithm

The natural logarithm, often denoted \( \ln \), is a logarithm to the base \( e \), where \( e \) is approximately 2.718. The natural logarithm is particularly useful in solving equations involving exponential decay because it effectively "undoes" the exponential function.
In the context of exponential decay, we often encounter equations like \( e^{-kt} = \frac{1}{2} \). To solve for \( t \), the natural logarithm helps to isolate \( t \).
Steps using natural logarithm:

• Start with \( \frac{1}{2} = e^{-kt} \).
• Apply the \( \ln \) to both sides: \( \ln(\frac{1}{2}) = -kt \).
• Solve for \( t \): \( t = \frac{\ln(\frac{1}{2})}{-k} \).

This process allows us to find the time at which a substance, like iodine-125, reaches half its original quantity.

Iodine-125

Iodine-125 is a radioactive isotope frequently used in medical treatments and diagnostics, particularly in cancer therapy. Its significance lies in its predictable decay pattern, which is essential for precise treatment planning.
As a radioactive substance, iodine-125 undergoes exponential decay. This means its quantity decreases exponentially over time, characterized by its half-life, which is the time it takes for half of the initial substance to decay.
Iodine-125 is particularly useful because:

• It emits low-energy radiation suited for targeting specific cells.
• Its predictable decay helps in planning therapeutic doses.
• The half-life allows for controlled, steady decay over time, maximizing treatment effectiveness.

Understanding these properties is key to harnessing iodine-125's benefits in medical treatments efficiently.