Problem 31 - College Algebra with Corequisite Support

Question

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

Step-by-Step Solution

Answer

It will take approximately 60 days for half of the Iodine-125 to decay.

Step 1: Understand the Decay Process

The problem involves radioactive decay, which follows an exponential decay model. The amount left after a certain time can be described by the formula: \( N(t) = N_0 e^{-kt} \), where \( N_0 \) is the initial amount, \( k \) is the decay constant, and \( t \) is the time.

Step 2: Identify Given Information

We are given the initial amount of Iodine-125, \( N_0 = 0.5 \) grams, and the decay rate is \( 1.15\% \) per day. We want to find the time \( t \) when half of the initial amount remains, meaning \( N(t) = 0.25 \) grams.

Step 3: Convert Percentage to Decay Constant

The decay rate is \( 1.15\% \) per day, which means \( k = 0.0115 \) when converted to a decimal to use in the formula.

Step 4: Apply the Exponential Decay Formula

Set the equation \( N(t) = N_0 e^{-kt} \) for half-life: \( 0.25 = 0.5 e^{-0.0115t} \). Simplify to get \( 0.5 = e^{-0.0115t} \).

Step 5: Solve for Time (t)

Take the natural logarithm of both sides to solve for \( t \). This gives \( \ln(0.5) = -0.0115t \). Then, \( t = \frac{\ln(0.5)}{-0.0115} \).

Step 6: Calculate the Time

Using a calculator, compute \( t = \frac{-0.6931}{-0.0115} \), which approximately equals 60.27 days. Since the problem asks for the nearest day, round 60.27 to 60.

Key Concepts

Radioactive Decay

Radioactive decay is a process by which an unstable atomic nucleus loses energy by emitting radiation. It occurs naturally in many elements, and it is unpredictable for any single atom when it will happen. However, when you have a large number of atoms, the decay process becomes predictable as an average rate, which can be described mathematically.

This phenomenon follows an exponential decay model, which means that the quantity of the radioactive substance decreases over time at a rate proportional to its current value. You can use the formula: \[ N(t) = N_0 e^{-kt} \] where:

• \(N(t)\) is the quantity at time \(t\).
• \(N_0\) is the initial amount of the substance.
• \(e\) is the base of the natural logarithm (approximately equal to 2.71828).
• \(k\) is the decay constant.

Understanding the concept of radioactive decay is crucial in various fields such as physics, medicine, and environmental science.

It helps in predicting how long it will take for a substance to reduce to a certain amount and is especially significant in fields that use radiometric dating or manage radioactive materials.

Half-life Calculation

Half-life is a term used to describe the time required for one-half of a radioactive substance to decay. In simple terms, it is the period for reducing the original amount of the substance to 50% of its initial value. This concept is pivotal in understanding how quickly or slowly a substance undergoes decay.

For example, in the given exercise, the half-life of Iodine-125 was calculated based on its decay rate. By setting the amount at half of the original in the exponential decay formula, the half-life can be derived. The resulting equation is represented as:\[ N_0 imes rac{1}{2} = N_0 e^{-kt_{1/2}} \] where \( t_{1/2} \) is the half-life.

Through rearranging and computing the above gives:\[ t_{1/2} = rac{ ext{ln}(0.5)}{-k} \]

Understanding the half-life helps in applications such as medicine (for the decay of isotopes used in treatment), archaeology (carbon dating), and nuclear power management.

Decay Constant

The decay constant symbolizes the rate at which a radioactive substance decays. It provides a measure for how fast the decay process occurs, often specific to each radioactive isotope.

In mathematics, the decay constant \(k\) is used in the exponential decay formula and is typically expressed in inverse units of time. It is related to the percent decay rate given, and can be converted using:\[ k = \frac{ ext{Decay Rate}}{100} \]

For Iodine-125, with a decay rate of 1.15% per day, calculating the decay constant involves converting this percentage to a decimal, resulting in \(k = 0.0115\).

The decay constant provides crucial information for computational predictions concerning the decay of radioactive materials. By knowing \(k\), one can accurately model the exponential decay, forecast the remaining amount over time, and calculate the half-life of the substance.