Question

Devise an exponential decay function that fits the following data; then answer the accompanying questions. Be sure to identify the reference point \((t=0)\) and units of time. Roughly 12,000 Americans are diagnosed with thyroid cancer every year, which accounts for \(1 \%\) of all cancer cases. It occurs in women three times as frequently as in men. Fortunately, thyroid cancer can be treated successfully in many cases with radioactive iodine, or I-131. This unstable form of iodine has a half-life of 8 days and is given in small doses measured in millicuries. a. Suppose a patient is given an initial dose of 100 millicuries. Find the function that gives the amount of I-131 in the body after \(t \geq 0\) days. b. How long does it take for the amount of I-131 to reach 10\% of the initial dose? c. Finding the initial dose to give a particular patient is a critical calculation. How does the time to reach \(10 \%\) of the initial dose change if the initial dose is increased by \(5 \% ?\)

Short answer

Answer: The time it takes to reach 10% of the initial dose does not change when the initial dose is increased by 5%. In both cases, it takes approximately 26.57 days to reach 10% of the initial dose.

Step-by-step solution

a. Find the function for the amount of I-131 in the body

Use the exponential decay formula: \(A(t) = A_0 \cdot (1/2)^{\frac{t}{h}}\), where \(A_0 = 100 \text{ millicuries}\) and \(h = 8 \text{ days}\). So, the function is: \(A(t) = 100 \cdot (1/2)^{\frac{t}{8}}\)

b. Find the time to reach 10% of the initial dose

To find the time when the amount of I-131 reaches 10% of the initial dose, we need to find \(t\) such that \(A(t) = 0.1 \cdot A_0\) (10% of the initial dose). So we have the equation: \(100 \cdot (1/2)^{\frac{t}{8}} = 0.1 \cdot 100\) Divide both sides by 100: \((1/2)^{\frac{t}{8}} = 0.1\) Take the logarithm base 0.5 of both sides: \(\frac{t}{8} = \log_{0.5}0.1\) Finally, find the value of \(t\) by multiplying both sides by 8: \(t = 8\cdot \log_{0.5}0.1 \approx 26.57 \text{ days}\)

c. Increasing the initial dose by 5%

Let's find how the time to reach 10% of the initial dose changes if the initial dose is increased by 5%. Increasing the initial dose by 5% means a new initial amount \(A'_0=1.05\cdot100=105 \text{ millicuries}\). Now, we need to find \(t\) such that \(A'(t) = 0.1 \cdot A'_0\). Our new function is: \(A'(t) = 105 \cdot (1/2)^{\frac{t}{8}}\) Set up the equation: \(105 \cdot (1/2)^{\frac{t}{8}} = 0.1 \cdot 105\) Divide both sides by 105: \((1/2)^{\frac{t}{8}} = 0.1\) Take the logarithm base 0.5 of both sides: \(\frac{t}{8} = \log_{0.5}0.1\) Finally, find the value of \(t\) by multiplying both sides by 8: \(t = 8\cdot \log_{0.5}0.1 \approx 26.57 \text{ days}\) Both times to reach 10% of the initial doses are the same, approximately 26.57 days. When we change the initial dose, it doesn't affect the time it takes to reach 10% of the initial dose in this exponential decay problem.

Key concepts

Half-life

Half-life is a concept used in exponential decay, which describes the time it takes for a substance to decrease to half of its original amount. This idea is particularly important in nuclear physics and medicine, especially when dealing with substances like radioactive iodine, known as I-131. Understanding half-life helps doctors determine how long it will take for a radioactive treatment to diminish in the body, aiding in patient safety and treatment planning.
For I-131, the half-life is 8 days, meaning every 8 days, half of the radioactive iodine is metabolized or expelled from the body. This property allows physicians to plan treatments effectively, knowing exactly how long the substance remains active and to what degree.
The half-life formula used in the context of exponential decay is: \[ A(t) = A_0 \cdot \left( \frac{1}{2} \right)^{\frac{t}{h}} \]where:

• \(A(t)\) is the amount of the substance at time \(t\).
• \(A_0\) is the initial amount.
• \(h\) is the half-life period.

Radioactive Iodine

Radioactive iodine, particularly I-131, is a commonly used treatment for certain types of thyroid conditions, including thyroid cancer. It works by emitting radiation that destroys thyroid cells, helping to treat the disease. Once introduced into the body, the radioactive iodine steadily diminishes due to its radioactive decay process, governed by its half-life of 8 days.
Patients are typically administered small, calculated doses measured in millicuries to ensure effectiveness while minimizing risk. Understanding the decay process is vital for transiting to other phases of treatment or terminating it, as leftover radioactive material can be harmful. The body processes it naturally over time, reducing the radiation dose and limiting side effects.
In medical contexts, accurately predicting how long the iodine will remain active in the patient's system is crucial for achieving optimal therapeutic results, allowing doctors to make informed decisions about scheduling subsequent treatments or follow-up appointments.

Logarithmic Functions

Logarithmic functions are mathematical tools especially useful in calculating the time it takes for exponential decay to occur, like determining how long it takes for a radioactive material to decay to a particular amount. The logarithm helps solve equations that arise when dealing with exponential decay, where the rate of change is proportional to the current value.
Using the equation:\[(1/2)^{\frac{t}{8}} = 0.1\]we can apply logarithms to isolate \(t\). The logarithm base 0.5 is useful because it was derived from the half-life relationship, allowing us to solve for the time at which a substance reaches a specific fraction of its initial amount.
The process involves taking the logarithm of both sides:\[\frac{t}{8} = \log_{0.5}0.1\]providing a direct method to find \(t\). Logarithms translate complex exponential relationships into more manageable linear equations, making them essential in scientific and engineering calculations.

Exponential Functions

Exponential functions are central to understanding how quantities change over time in processes such as radioactive decay. They describe situations where the rate of change is proportional to the current amount, resulting in a rapid decrease or growth.For example, an exponential decay function for radioactive iodine can be expressed as:\[A(t) = A_0 \cdot (1/2)^{\frac{t}{8}}\]where:

• \(A(t)\) is the amount left after \(t\) days.
• \(A_0\) is the initial dose.
• \(1/2\) represents the halving from the half-life.

Exponential decay functions are helpful in predicting how a substance like I-131 depletes over time, an insight that is critical for medical treatments and safety protocols. They provide precise models for understanding biological and physical systems, enabling professionals to compute exact doses, anticipate results, and manage timelines effectively.