Problem 54 - Introductory and Intermediate Algebra for College Students 4th

Question

Use a calculator with a \(\overline{y^{x}}\) key or \(a \ \triangle\) key to solve. The 1986 explosion at the Chernobyl nuclear power plant in the former Soviet Union sent about 1000 kilograms of radioactive cesium-137 into the atmosphere. The function \(f(x)=1000(0.5)^{\frac{x}{30}}\) describes the amount, \(f(x),\) in kilograms, of cesium-137 remaining in Chernobyl \(x\) years after \(1986 .\) If even 100 kilograms of cesium- 137 remain in Chernobyl's atmosphere, the area is considered unsafe for human habitation. Find \(f(80)\) and determine if Chernobyl will be safe for human habitation by 2066

Step-by-Step Solution

Answer

Use a calculator to compute \(f(80)=1000(0.5)^{\frac{80}{30}}\). Then check whether this amount is less than 100 kilograms to conclude whether Chernobyl will be safe by 2066 or not.

Step 1: Understand the function

Given decay function is \(f(x)=1000(0.5)^{\frac{x}{30}}\). This function tells us the amount of cesium-137 remaining after \(x\) years. We need to find \(f(80)\) which means the amount of cesium-137 left 80 years after 1986.

Step 2: Calculate \(f(80)\)

Substituting \(x = 80\) into the function, we have: \(f(80)=1000(0.5)^{\frac{80}{30}}\). Use a calculator to perform this calculation.

Step 3: Evaluate Safety

If the result from step 2 is less than 100, Chernobyl will be safe for human habitation by 2066. Otherwise, it will still be considered unsafe.

Key Concepts

Radioactive Decay

Radioactive decay is a process where unstable atomic nuclei lose energy by emitting radiation. This process occurs naturally and results in the transformation of one element into another over time. In the context of exponential decay, the substance reduces its quantity by a consistent percentage over equal time intervals.
For cesium-137, an unstable isotope released during the Chernobyl disaster, it is essential to determine how quickly or slowly it decays. The decay is modeled by the equation \(f(x)=1000(0.5)^{\frac{x}{30}}\), which reflects the half-life of cesium-137. A half-life is the amount of time required for half of the radioactive atoms in a sample to decay. For cesium-137, the half-life dramatically affects how long areas remain hazardous.

• Half-life: The time it takes for half of the radioactive atoms to decay.
• Exponentially decreasing functions model the radioactive decay process.
• Safety evaluations depend on whether decayed amounts bring radiation below dangerous levels.

Understanding this function helps us predict the future safety of affected areas through science and mathematics.

Mathematical Functions

Mathematical functions describe relationships between quantities and are essential tools in scientific modeling. The function \(f(x)=1000(0.5)^{\frac{x}{30}}\) is a specific exponential decay function, meaning it decreases rapidly over time as represented by the base \(0.5\).
In this function:

• \(x\) represents the number of years since the Chernobyl disaster.
• \(1000\) represents the initial amount of cesium-137 released.
• The term \(0.5\) shows the percentage left after each half-life.
• The exponent \(\frac{x}{30}\) adjusts the decay rate to match the half-life of cesium-137.

Using these components, one can calculate the remaining radioactive material at any given time. With mathematical tools, we solve for specific values such as \(f(80)\) to determine cesium-137 levels decades after the incident.

Chernobyl Nuclear Disaster

The Chernobyl Nuclear Disaster in 1986 was one of the most catastrophic nuclear incidents in history. It occurred when a reactor at the Chernobyl Power Plant, located in the former Soviet Union's territory, exploded, releasing significant quantities of radioactive materials like cesium-137 into the atmosphere.
The resulting contamination had far-reaching effects on the environment and human health, leading to the evacuation and resettlement of people from the affected area. The legacy of this disaster lingers to this very day.
Key points about the Chernobyl disaster include:

• The explosion released about 1000 kilograms of radioactive cesium-137.
• The persistent radiation poses ongoing health risks to living organisms.
• Affected areas remain hazardous and require considerable time to become safe.

Mathematical models like exponential decay equations are crucial for understanding how this radioactive material decreases over time, allowing experts to evaluate when the area might become safe for human habitation.