Problem 62 - College Algebra with Corequisite Support

Question

A scientist begins with 100 milligrams of a radioactive substance that decays exponentially. After 35 hours, \(50 \mathrm{mg}\) of the substance remains. How many milligrams will remain after 54 hours?

Step-by-Step Solution

Answer

Approximately 37.61 mg will remain after 54 hours.

Step 1: Understanding Exponential Decay Formula

In problems involving exponential decay, the formula used is \( N(t) = N_0 e^{-kt} \), where \( N(t) \) is the quantity remaining after time \( t \), \( N_0 \) is the initial quantity, and \( k \) is the decay constant. We need to find \( k \) using the given information.

Step 2: Calculating the Decay Constant \( k \)

Given that \( N_0 = 100 \) mg and \( N(35) = 50 \) mg after 35 hours, we substitute these values into the exponential decay formula: \[ 50 = 100 e^{-35k} \]. Solving for \( k \), we get \[ e^{-35k} = \frac{1}{2} \], which simplifies to \[ -35k = \ln\left(\frac{1}{2}\right) \]. Thus, \[ k = -\frac{\ln\left(\frac{1}{2}\right)}{35} \].

Step 3: Substitute \( k \) to Find Remaining Substance After 54 Hours

With \( k \) calculated, use the formula \( N(t) = 100 e^{-kt} \), where \( t = 54 \). Substitute \( k \) and \( t \) into the equation: \[ N(54) = 100 e^{-\frac{\ln\left(\frac{1}{2}\right)}{35} \cdot 54} \]. This can be simplified using properties of exponents as \[ N(54) = 100 \left( \frac{1}{2} \right)^{\frac{54}{35}} \].

Step 4: Calculate the Remaining Milligrams

Using a calculator, evaluate \[ 100 \times \left( \frac{1}{2} \right)^{\frac{54}{35}} \] and find that \( N(54) \approx 37.61 \) mg. Thus, after 54 hours, approximately 37.61 milligrams of the substance will remain.

Key Concepts

Decay Constant

The decay constant, represented as \( k \), is a key component in the exponential decay formula. It helps determine the rate at which a radioactive substance diminishes over time. Generally, decay constant is unique to each substance, allowing us to quantify how quickly or slowly a substance breaks down.
For example, the process to find the decay constant involves using known values of the substance. When you know how much substance remains after a specific period, you can apply the exponential decay formula:

• \( N(t) = N_0 e^{-kt} \), where:
• \( N(t) \) is the amount remaining after time \( t \)
• \( N_0 \) is the initial quantity
• \( k \) is the decay constant

With this equation, solve for \( k \) by rearranging terms. In our example, this was done by knowing \( N_0 = 100 \) mg and \( N(35) = 50 \) mg. Substitution and solving will lead you to find \( k \) accurately. It's a measure enabling scientists to predict future quantities of the substance remaining over time.

Radioactive Substance

A radioactive substance is a material that naturally breaks down over time, emitting radiation through a process called radioactive decay. This spontaneous decay changes the composition of the substance, reducing its original mass.
When dealing with radioactive substances, understanding their behavior is crucial for applications like medical treatments, carbon dating, and nuclear power. Characteristics that define radioactive substances include:

• Decay over time, changing to different elements or isotopes
• Emissions of alpha, beta, or gamma radiation
• Unpredictable decay times for individual atoms but predictable for large groups

The rate at which a particular radioactive substance decays is an intrinsic property, often influenced by external conditions like pressure and temperature. The behavior of these substances is characterized by their constant and natural process, thus needing the decay constant to help in calculations.

Half-life Calculation

Half-life is a crucial concept when dealing with radioactive decay, referring to the time taken for half of a radioactive substance to decay. This allows scientists to predict how long a substance will remain active or effective. Calculating half-life involves understanding that the substance's mass reduces by half over each half-life period.
To calculate half-life, you can use the decay constant \( k \):

• The relationship is given by \( t_{1/2} = \frac{\ln(2)}{k} \)

In the exercise problem, knowing that the radioactivity halved in 35 hours allows for the calculation of the decay constant, which directly leads to understanding the scaling of decay over time. This consistent period of loss exponentially affects the remaining quantities, highlighting how quickly substances decay relative to others. Understanding the half-life helps in various fields such as medicine (for calculating dosages of radioactive tracers) and environmental science (in nuclear waste management).