Problem 19 - Applied Calculus

Question

One of the main contaminants of a nuclear accident, such as that at Chernobyl, is strontium- \(90,\) which decays exponentially at a rate of approximately \(2.5 \%\) per year. (a) Write the percent of strontium- 90 remaining. \(P\), as a function of years, \(t,\) since the nuclear accident. IHint: \(100 \%\) of the contaminant remains at \(t=0.1\) (b) Graph \(P\) against \(t\) (c) Estimate the half-life of strontium- 90 (d) After the Chernobyl disaster, it was predicted that the region would not be safe for human habitation for 100 years. Estimate the percent of original strontium-90 remaining at this time.

Step-by-Step Solution

Answer

(a) \( P(t) = 100 e^{-0.025t} \); (c) Half-life is about 27.7 years; (d) ~8.2% remains after 100 years.

Step 1: Determine The Exponential Decay Function

Strontium-90 decays at a rate of 2.5% per year. The exponential decay function is of the form \( P(t) = P_0 e^{rt} \), where \( r \) is the decay rate and \( P_0 \) is the initial quantity. Substituting \( r = -0.025 \) and \( P_0 = 100 \), the function becomes:\[ P(t) = 100 e^{-0.025t} \]

Step 2: Graph the Decay Function

To graph \( P(t) = 100 e^{-0.025t} \), choose a range of \( t \) values, such as from 0 to 100. Calculate \( P(t) \) for these \( t \)-values and plot them. The graph will show an exponentially decaying curve starting from 100% at \( t = 0 \).

Step 3: Calculate the Half-Life

The half-life is the time it takes for the substance to reduce to 50% of its initial value. Set \( P(t) = 50 \) in the decay equation \( 100 e^{-0.025t} = 50 \). Solve for \( t \):\[ e^{-0.025t} = 0.5 \]\[ -0.025t = \, \ln(0.5) \]\[ t = \frac{\ln(0.5)}{-0.025} \approx 27.7 \text{ years} \]

Step 4: Estimate The Remaining Strontium-90 After 100 Years

Insert \( t = 100 \) into the decay function:\[ P(100) = 100 e^{-0.025 \times 100} \]\[ P(100) = 100 e^{-2.5} \approx 8.2 \]This means about 8.2% of the original strontium-90 remains.

Key Concepts

Half-Life Calculation

The concept of half-life is crucial when dealing with radioactive substances, such as strontium-90. Half-life refers to the time it takes for half of the radioactive atoms in a sample to decay. Identifying this helps us understand how quickly a substance like strontium-90 will decay over time.

To calculate the half-life of strontium-90, we use the exponential decay formula: \[ P(t) = P_0 e^{rt} \]Here, the initial quantity \( P_0 = 100 \) and the decay rate \( r = -0.025 \). Setting \( P(t) = 50 \) helps us find when the substance has reduced to half. The equation becomes:\[ 100 e^{-0.025t} = 50 \]Solving for \( t \), you rearrange and compute:\[ e^{-0.025t} = 0.5 \]\[ -0.025t = \ln(0.5) \]\[ t = \frac{\ln(0.5)}{-0.025} \] This calculation yields approximately \( 27.7 \) years, which is the half-life of strontium-90. Understanding half-life enables predicting how long a radioactive material remains hazardous.

Decay Function Graphing

Graphing the decay function provides a visual representation of how the quantity of strontium-90 decreases over time. It's a critical step for understanding exponential decay.

To plot this, we use the decay function:\[ P(t) = 100 e^{-0.025t} \]We identify a range of \( t \) values, like 0 to 100 years, and calculate \( P(t) \) for each \( t \).- At \( t = 0 \), \( P(0) = 100 \)%, representing the initial quantity.- As \( t \) increases, \( P(t) \) decreases exponentially.This results in a curve starting at 100% and moving downward as \( t \) increases. The curve almost approaches zero but never quite reaches it. This graph indicates how quickly or slowly radioactive material diminishes over time.

Interpreting this graph can help determine when an area affected by radioactive contamination might become safer depending on the percentage of strontium-90 remaining.

Radioactive Contamination

Radioactive contamination refers to the presence of radioactive substances like strontium-90 in the environment.

Contamination often occurs after accidents such as Chernobyl, spreading radiation through air, water, and soil. Understanding the rate of decay helps estimate the time required for an area to become safe. - After a nuclear accident, it's essential to assess how long an area will remain dangerous. - Strontium-90's half-life shows it remains hazardous for several decades. - Calculating the percentage of remaining radioactive material over time aids safety predictions. In the case of Chernobyl, within 100 years, only about 8.2% of the original strontium-90 remains. While significantly reduced, this still needs careful consideration for safety measures and potential human habitation.

Long-term monitoring and management strategies are crucial to mitigate the impact of radioactive contamination on the environment and public health. Understanding decay patterns assists policymakers in planning prompt and effective responses to these environmental hazards.