Problem 51 - Precalculus

Question

A radioactive isotope of selenium, 'sse, which is used in medical imaging of the pancreas, has a half-life of 119.77 days. If 200 milligrams are given to a patient, how many milligrams are left after 30 days?

Step-by-Step Solution

Answer

Approximately 168 milligrams remain after 30 days.

Step 1: Determine the Decay Constant

The half-life of the isotope is 119.77 days. The decay constant \(k\) can be calculated using the formula for half-life: \(k = \frac{\ln(2)}{t_{1/2}}\), where \(t_{1/2} = 119.77\) days. Substitute the given values to find \(k\).\[ k = \frac{\ln(2)}{119.77} \approx 0.005788 \text{ days}^{-1} \]

Step 2: Use the Exponential Decay Formula

The remaining amount \(A(t)\) of a substance at time \(t\) with initial amount \(A_0\) can be calculated using the formula: \(A(t) = A_0 e^{-kt}\), where \(A_0 = 200\) mg and \(t = 30\) days.

Step 3: Substitute the Known Values

Substitute \(A_0 = 200\), \(k = 0.005788\), and \(t = 30\) into the exponential decay formula:\[ A(30) = 200 e^{-0.005788 \times 30} \]

Step 4: Calculate the Remaining Amount

First, compute the exponent: \[ -0.005788 \times 30 = -0.17364 \]Then substitute and solve for \(A(30)\):\[ A(30) = 200 e^{-0.17364} \approx 200 \times 0.8400 \approx 168 \] milligrams.

Key Concepts

Half-life

The half-life of a radioactive isotope is a key concept when discussing exponential decay. It is defined as the time required for half of a sample of a radioactive substance to decay. In simple terms, if you have a certain amount of a radioactive material, the half-life is how long it takes for just half of it to turn into something else due to decay.
This time frame depends on the specific isotope and can range from fractions of a second to millions of years. Using the half-life helps us understand how quickly or slowly a radioactive substance reduces in quantity over time.
For example, in the given exercise, the half-life of the radioactive isotope of selenium is 119.77 days. This means that every 119.77 days, the amount of selenium is reduced by half.

Radioactive Isotope

A radioactive isotope, also known as a radioisotope, is an element with an unstable nucleus that decays over time. This decay process leads to the emission of radiation, such as alpha particles, beta particles, or gamma rays.
Radioactive isotopes occur naturally but can also be produced artificially. They are used in various fields including medicine, research, and industry. In medicine, radioisotopes are used in imaging and treatment because their specific decay properties allow them to be tracked or to destroy cancer cells.
In the problem you encounter, the radioactive isotope of selenium is used in medical imaging to help visualize the pancreas. Its decay properties are critical as they determine how long the isotope remains effective and safe within a patient's body.

Exponential Decay Formula

The exponential decay formula is essential for calculating how much of a radioactive isotope remains after a period of time. The formula is:

• The initial amount of the substance, represented as \( A_0 \).
• The decay constant, represented by \( k \), which is derived from the half-life.
• The time elapsed, represented by \( t \).

The formula itself is: \( A(t) = A_0 e^{-kt} \). Here, \( A(t) \) indicates the remaining amount at time \( t \). This formula is derived from the natural exponential function due to the continuously proportional decrease of the substance over time.
Using this formula allows us to predict how much of the radioisotope will be left after a given time period. In the exercise, by substituting known values such as the initial amount, the decay constant, and the elapsed time of 30 days, we can calculate that 168 milligrams of selenium remain after this period.