Problem 53 - College Algebra with Corequisite Support

Question

A doctor injects a patient with 13 milligrams of radioactive dye that decays exponentially. After 12 minutes, there are 4.75 milligrams of dye remaining in the patient's system. Which is an appropriate model for this situation? (a) \(f(t)=13(0.0805)^{t}\) (b) \(f(t)=13 e^{0.9195 t}\) (c) \(f(t)=13 e^{(-0.0839 t)}\) (d) \(f(t)=\frac{4.75}{1+13 e^{-0.83925 t}}\)

Step-by-Step Solution

Answer

The appropriate model is (c) \(f(t) = 13 e^{-0.0839 t}\).

Step 1: Understanding Exponential Decay

The problem involves exponential decay, where the quantity of radioactive dye decreases over time. Exponential decay can be modeled by the formula: \( f(t) = A e^{-kt} \), where \( A \) is the initial amount, \( k \) is the decay constant, and \( t \) is time in appropriate units.

Step 2: Setting Up the Equation

We know from the problem that initially, \( A = 13 \) milligrams, and after 12 minutes, \( f(12) = 4.75 \) milligrams. We can use this information to determine the decay constant \( k \).

Step 3: Using the Decay Formula

Plug the known values into the decay formula: \( f(12) = Ae^{-kt} \). So, \( 4.75 = 13 e^{-12k} \).

Step 4: Solving for the Decay Constant (k)

Rearrange the equation to solve for \( k \): \[4.75 = 13 e^{-12k} \\frac{4.75}{13} = e^{-12k} \\ln\left(\frac{4.75}{13}\right) = -12k \k = -\frac{\ln\left(\frac{4.75}{13}\right)}{12}\]

Step 5: Calculating the Decay Constant

Using a calculator to find \( \ln\left(\frac{4.75}{13}\right) \), we find that \( k \approx 0.0839 \).

Step 6: Matching the Model to Options

Now that we know \( k \approx 0.0839 \), substitute \( k \) into the decay model: \( f(t) = 13 e^{-0.0839t} \). Comparing to the given options, it matches option (c): \( f(t) = 13 e^{-0.0839 t} \).

Key Concepts

Radioactive Dye

Radioactive dye is a substance used in medical imaging to enhance the visibility of structures or fluids within the body. It contains a radioactive element that emits radiation, which can be tracked by special imaging equipment. This radiation helps doctors see detailed images of the targeted areas in a non-invasive way. The dye is typically injected into the bloodstream, and its path through the body can provide valuable information for diagnosing and monitoring various medical conditions. The amount of radioactive dye diminishes over time as it decays, which is crucial for calculating its rate of decay when modeling its behavior mathematically.

Decay Constant

The decay constant, often denoted by the symbol \( k \), is a crucial component in the mathematical modeling of exponential decay. It describes the rate at which a material or substance, like radioactive dye, diminishes over time.

• The value of \( k \) is derived from the natural logarithm of the ratio of the remaining amount of substance to its initial quantity.
• This constant is specific to the substance being studied and its environmental conditions.
• In practical terms, the decay constant helps us understand how quickly the substance is losing its original mass.

For example, in this exercise, the decay constant is found by solving the equation \( - \frac{\ln\left(\frac{4.75}{13}\right)}{12} \), yielding a value of approximately \( 0.0839 \). This quantifies how fast the radioactive dye is decaying in the given scenario.

Exponential Model

An exponential model is a mathematical representation used to describe phenomena that change at rates proportional to their current value. This is especially relevant in contexts like radioactive decay where substances decrease in quantity swiftly over time. The general form of an exponential decay model is given by:\[ f(t) = A e^{-kt} \]- Here, \( A \) represents the initial quantity of the substance.- \( e \) is the base of the natural logarithm (approximately 2.71828).- \( k \) is the decay constant.- \( t \) is time.This decay formula helps predict the future amount of a substance given its initial amount and decay rate. In our case, the exponential model used was \( f(t) = 13e^{-0.0839t} \), which effectively modeled the decay of the radioactive dye in the patient's system.

Problem Solving Steps

Solving an exponential decay problem involves clear, logical steps. This ensures accuracy and understanding.

• Identify the initial conditions: Start by noting the initial amount and the remaining amount after a certain time.
• Set up the decay equation: Use the given information to plug values into the formula \( f(t) = A e^{-kt} \).
• Solve for the decay constant: Use algebraic manipulation and natural logarithms to isolate and solve for \( k \).
• Match findings to provided models: Compare your model with the options given to select the correct or most appropriate one.

By following these steps, one can efficiently determine the appropriate mathematical model for scenarios involving exponential decay, similar to selecting option (c) in this exercise, \( f(t) = 13 e^{-0.0839 t} \).