Problem 22 - Algebra and Trigonometry

Question

Radioactive Decay Doctors use radioactive iodine 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; (b) approx. 1.0548 grams after 20 days.

Step 1: Understanding the Initial Mass

To find the mass of iodine at time \(t=0\), we substitute \(t=0\) into the function \(m(t)=6 e^{-0.087 t}\). This step evaluates the initial condition of the function.

Step 2: Calculating the Initial Mass

Substitute \(t=0\) into \(m(t)=6 e^{-0.087 t}\):\[m(0) = 6 e^{-0.087 \times 0} = 6 e^0 = 6 \times 1 = 6\text{ grams}\]}The mass at time \(t=0\) is 6 grams.

Step 3: Understanding the Function for Time t=20 days

Next, we find the mass after 20 days. For this, substitute \(t=20\) into the function \(m(t)=6 e^{-0.087 t}\).

Step 4: Calculating the Mass at t=20 Days

Substitute \(t=20\) into the function:\[m(20) = 6 e^{-0.087 \times 20} = 6 e^{-1.74}\]Calculate \(e^{-1.74}\) using a calculator, which is approximately 0.1758:\[m(20) = 6 \times 0.1758 \approx 1.0548\text{ grams}\]So, the mass remaining after 20 days is approximately 1.0548 grams.

Key Concepts

Exponential Decay

Exponential decay is a process where a quantity decreases at a rate proportional to its current value. In the context of radioactive decay, this concept is pivotal for understanding how substances like radioactive iodine transform over time. When we talk about exponential decay, we refer to a continuous decrease - much like how a melting ice cube shrinks in size over time. The decay is never sudden. Instead, it's gradual and predictable.
This predictability is a crucial concept because it allows us to use mathematical functions to forecast the amount of substance left after a certain period. For instance, in the equation given in the problem, you can see the exponential term, represented by the natural exponential base, \(e\), and an exponent that contains \(-0.087t\). This negative rate exponent dictates how quickly the iodine's mass reduces over time.

• The formula for exponential decay is often written as \(N(t) = N_0 e^{-kt}\), where \(N_0\) is the initial quantity, \(k\) is the decay constant, and \(t\) is time.

Understanding exponential decay helps us predict how much of a radioactive substance will remain at any given time, aiding in activities like medical diagnostics or safety procedures.

Function Evaluation

Function evaluation involves replacing the variable in a function with a given number and then performing the necessary calculations. It is the essence of using mathematical equations to solve practical problems. In the provided exercise, the function \(m(t) = 6e^{-0.087t}\) is used to model the radioactive decay of iodine.
To evaluate this function, one must understand that each part of the function has a role. The number \(6\) signifies the initial mass, while \(e^{-0.087t}\) describes how this mass decreases over time.

• When evaluating for specific times, simple substitution into the equation allows us to find the remaining iodine, making it an essential skill in applied sciences.
• For instance, substituting \(t=0\) yields the initial mass, and \(t=20\) tells us the mass after 20 days.

Function evaluation is like filling in the blanks in a storybook. By inputting different times, we create a narrative of the iodine's decay over time, offering insights into the molecule's behavior at each stage.

Initial Mass

The initial mass is the starting amount of a substance before any change, decay, or transformation occurs. For this specific iodine sample, we were tasked to find the initial mass at \(t=0\), which essentially means 'at the beginning'.
By substituting \(t=0\) in the decay formula \(m(t) = 6e^{-0.087t}\), we determine the initial mass. Since the exponential term \(e^0\) equals 1, the initial mass is simply \(6\) grams. This is the known quantity that helps establish the decay model.

• The initial mass comes into play as the baseline from which all decay calculations begin, essentially anchoring our exponential decay curve.

Understanding the initial mass is like identifying the starting line in a race; it's where you set the scene and begin measuring all subsequent changes.

Calculating Remaining Mass

To calculate the remaining mass at any given time during a decay process, we substitute the desired time into the function. In our example with iodine, this involves inserting \(t=20\) into the function \(m(t) = 6e^{-0.087t}\) to find the remaining mass after 20 days.
This calculation makes use of the negative exponent, reflecting how rapidly the material is decaying. Highlights of this process include:

• Substitute the time into \(t\) in the decay formula.
• Use a calculator to evaluate the exponential term, \(e^{-1.74}\), which approximates to \(0.1758\).
• Multiply this outcome by the initial mass \(6\) to obtain about \(1.0548\) grams left after 20 days.

Calculating the remaining mass allows us to determine precisely how much of the radioactive iodine is still viable for medical tracers, which could affect diagnostic outcomes.

Half-Life

The half-life of a substance is the time it takes for half of it to decay. It is a critical concept in radioactive decay because it offers a simple metric for describing the rate of decay. Though the term 'half-life' wasn't directly calculated in the problem, we can deduce information about it from the decay constant \(0.087\).
To calculate the half-life of a radioactive substance, use the formula \(t_{1/2} = \frac{ln(2)}{k}\), where \(k\) is the decay constant. By substituting \(k=0.087\), we find:

• \[t_{1/2} = \frac{ln(2)}{0.087} \approx 7.97\text{ days}\]

This calculation indicates the time required for half the iodine to decay, providing a crucial timeline for predicting when a sample becomes too weak to use effectively in tracers or other applications. Understanding half-life facilitates decision-making in storage, usage, and replacement of radioactive substances.