Problem 18 - Applied Calculus

Question

The half-life of a radioactive substance is 12 days. There are 10.32 grams initially. (a) Write an equation for the amount, \(A\), of the substance as a function of time. (b) When is the substance reduced to 1 gram?

Step-by-Step Solution

Answer

(a) \( A(t) = 10.32 \left(\frac{1}{2}\right)^{\frac{t}{12}} \); (b) It takes about 40.38 days.

Step 1: Understand the Half-Life Formula

For radioactive decay, the amount of substance remaining can be calculated using the half-life formula. The equation is given by:\[ A(t) = A_0 \left(\frac{1}{2}\right)^{\frac{t}{T}} \]where \( A(t) \) is the amount remaining after time \( t \), \( A_0 \) is the initial amount, and \( T \) is the half-life of the substance. In this problem, \( A_0 = 10.32 \) grams, and \( T = 12 \) days.

Step 2: Write the Function for Amount Over Time

Substitute the given values into the half-life equation. The function becomes:\[ A(t) = 10.32 \left(\frac{1}{2}\right)^{\frac{t}{12}} \]This equation represents the amount of the substance as a function of time \( t \).

Step 3: Set Up Equation for Remaining Amount

We need to find the time \( t \) at which the amount of substance is reduced to 1 gram. Set \( A(t) = 1 \) and use the equation:\[ 1 = 10.32 \left(\frac{1}{2}\right)^{\frac{t}{12}} \]

Step 4: Simplify and Solve for Time t

Rearrange the equation to solve for \( t \):\[ \frac{1}{10.32} = \left(\frac{1}{2}\right)^{\frac{t}{12}} \]Take the logarithm of both sides:\[ \log\left(\frac{1}{10.32}\right) = \frac{t}{12} \cdot \log\left(\frac{1}{2}\right) \]Solve for \( t \):\[ t = 12 \cdot \frac{\log\left(\frac{1}{10.32}\right)}{\log\left(\frac{1}{2}\right)} \]

Step 5: Calculate the Value of Time t

Compute the value using a calculator:\[ t \approx 12 \cdot \frac{-1.013}{-0.301} \approx 12 \cdot 3.365 \approx 40.38 \]Therefore, the time at which the substance is reduced to 1 gram is approximately 40.38 days.

Key Concepts

Half-life

The concept of half-life is crucial in understanding radioactive decay. It refers to the amount of time it takes for half of a given quantity of a radioactive substance to decay. This property is unique to each material, based on its nuclear composition.
It doesn't matter how much of the substance there is initially; after one half-life, only half of it remains. For example, if a substance has a half-life of 12 days, after 12 days only 50% will remain. After another 12 days (a total of 24 days), only 25% of the original amount remains, and so on.
This predictable pattern of decay allows scientists to use half-life to calculate the remaining quantity of a substance at any given time. It is a fundamental concept in fields like archaeology, medicine, and nuclear physics, enabling us to date ancient artifacts or understand how quickly a radioactive treatment will diminish in efficacy.

Amount of Substance Over Time

Calculating the amount of a radioactive substance over time involves tracking how much of the original material remains after a period. In the context of radioactive decay, this is often done using the half-life formula.
The equation to understand is:

• \( A(t) = A_0 \left(\frac{1}{2}\right)^{\frac{t}{T}} \)

Where:

• \( A(t) \) is the amount remaining after time \( t \)
• \( A_0 \) is the initial amount of the substance
• \( T \) is the half-life of the substance

This formula provides a clear pathway to determine how much of the substance remains after a specific time period. Let's say you start with 10.32 grams of a material with a half-life of 12 days, after which you can use the formula to determine how much remains at any given point by substituting the appropriate values.
This process is not only important in scientific measurements but also in practical applications like determining the safety of long-term exposure to radioactive materials. Understanding this formula allows us to plan effectively for safe interactions with these substances.

Exponential Decay Function

The exponential decay function is a mathematical representation of how quantities decrease over time. In the case of radioactive decay, it reflects how the amount of a substance diminishes at an exponential rate, dictated by its half-life.
Viewed graphically, the decay function forms a curved, downward-sloping line, steepest near the beginning where the substance reduces rapidly. As time progresses, the curve flattens, indicating a slower rate of decay.
Mathematically, each successive time period sees a consistent proportional reduction in substance, exemplified by:

• \( A(t) = A_0 \left(\frac{1}{2}\right)^{\frac{t}{T}} \)

This formula is a typical expression of exponential decay, showing how a fixed percentage of the total is reduced per unit of time. Thanks to its predictability, exponential decay functions are critical tools in various scientific fields, enabling accurate modeling of scenarios ranging from radioactive decay to depreciation in finance.
Understanding exponential functions helps us appreciate the predictable yet profound ways in which materials and values change over time, making it a fundamental part of both science and mathematics education.