Problem 66 - Algebra and Trigonometry

Question

Radioactive Decay Radioactive iodine is used by doctors as a tracer in diagnosing certain thyroid gland disorders. This type of iodine decays in such a way that the mass remaining after \(t\) days is given by the function $$ m(t)=6 e^{-0.087 t} $$ where \(m(t)\) is measured in grams. (a) Find the mass at time \(t=0\) (b) How much of the mass remains after 20 days?

Step-by-Step Solution

Answer

(a) 6 grams at \(t=0\). (b) Approximately 1.05 grams remain after 20 days.

Step 1: Evaluate Mass at Time Zero

To find the mass at time \( t = 0 \), we substitute \( t = 0 \) into the function \( m(t) = 6e^{-0.087t} \). This results in \( m(0) = 6e^{-0.087 \times 0} = 6e^{0} = 6 \text{ grams} \) because \( e^{0} = 1 \).

Step 2: Evaluate Mass After 20 Days

To determine how much of the mass remains after 20 days, substitute \( t = 20 \) into the function \( m(t) = 6e^{-0.087t} \). This gives \( m(20) = 6e^{-0.087 \times 20} = 6e^{-1.74} \). Calculate \( e^{-1.74} \) using a calculator, which is approximately \( 0.175 \). Therefore, \( m(20) \approx 6 \times 0.175 \approx 1.05 \text{ grams} \).

Key Concepts

Exponential Functions

Exponential functions are a type of mathematical function that describes a wide range of phenomena in the real world, especially in fields such as biology, physics, and economics. An exponential function is typically expressed in the form \( f(x) = a e^{bx} \), where:

• \(a\) is a constant representing the initial amount or size.
• \(e\) is the base of the natural logarithm, approximately equal to 2.71828.
• \(b\) is a constant that affects the rate of growth or decay.

These functions are particularly useful in modeling processes that grow or decay at rates proportional to their current value. For example, population growth, compound interest, and radioactive decay are modeled using exponential functions. In general, if \(b > 0\), the function is increasing, and if \(b < 0\), it is decreasing. In the context of radioactive decay, the function shows how the mass of a substance decreases over time.

Radioactive Iodine

Radioactive iodine, also known as iodine-131, is commonly used in medical applications, particularly in diagnosing and treating thyroid conditions. It is a radioactive isotope of iodine, meaning it emits radiation as it decays. It is used as a tracer in nuclear medicine because of its ability to concentrate in the thyroid gland. This characteristic makes it invaluable for:

• Diagnosing thyroid disorders such as hyperthyroidism and thyroid cancer.
• Performing therapies that require targeted radiation, as it selectively destroys thyroid cells.

The use of exponential decay functions in this context helps in understanding how the amount of radioactive iodine diminishes over time, providing insights into timing for safe and effective medical interventions.

Mass Calculation

Mass calculation in radioactive decay involves determining the remaining quantity of a radioactive substance after a given period. This is achieved using the exponential decay function \(m(t) = 6e^{-0.087t}\), where \(m(t)\) represents the mass at time \(t\) in grams. Understanding mass calculation is crucial because:

• It allows prediction of the remaining amount of radioactive material after a specific time.
• It is essential for ensuring patient safety in medical treatments, as the precise dosage of radioactive iodine is critical.

For example, initially, at time \(t = 0\), the mass is 6 grams, because \(e^{0} = 1\). At \(t = 20\) days, the mass is calculated to be approximately 1.05 grams, demonstrating how quickly radioactive iodine diminishes, which is essential for planning treatment schedules.

Decay Rate

The decay rate in an exponential function represents how quickly a substance decreases in amount over time. It is critical in the context of radioactive decay to ensure accurate predictions about the remaining substance. The rate of decay for radioactive iodine is denoted by the constant \(-0.087\) in the function \(m(t) = 6e^{-0.087t}\). Here, the decay rate explains:

• How fast the radioactive substance loses its mass.
• Helps in scheduling the administration of doses in therapeutic treatments.

An understanding of the decay rate allows healthcare professionals to calculate the half-life of the substance, which is the time it takes for half of the radioactive iodine to decay. This knowledge is critical for determining how long a patient needs to be monitored or treated, ensuring both efficacy and safety in medical applications.