Problem 38 - Algebra A Combined Function

Question

Solve. Unless otherwise indicated, round results to one decimal place. The nuclear waste from an atomic energy plant decays at a rate of \(3 \%\) each century. If 150 pounds of nuclear waste is disposed of, how much of it will still remain after 10 centuries? Use \(y=150(0.97)^{x}\), and let \(x\) be 10

Step-by-Step Solution

Answer

110.6 pounds of nuclear waste will remain after 10 centuries.

Step 1: Identify the Variables

We are given the formula \(y = 150(0.97)^x\) where \(y\) is the amount of nuclear waste remaining after \(x\) centuries. In this problem, \(x = 10\) because we are asked to calculate for 10 centuries.

Step 2: Substitute the Value of x

Substitute \(x = 10\) into the formula. The equation becomes: \[ y = 150(0.97)^{10} \]

Step 3: Calculate the Exponent

Calculate \((0.97)^{10}\).Using a calculator, \((0.97)^{10} \approx 0.7374\).

Step 4: Multiply by Initial Amount

Substitute \(0.7374\) back into the equation and multiply by 150.Calculate: \[ y = 150 \times 0.7374 \approx 110.6 \]

Step 5: Conclude with Final Answer

After calculating, we find that approximately 110.6 pounds of nuclear waste remains after 10 centuries.

Key Concepts

Nuclear Waste

Nuclear waste is a byproduct of nuclear reactions, typically derived from atomic energy plants or other nuclear applications. Its disposal and management pose a significant challenge due to its radioactive nature and potential detrimental impacts on the environment and human health. This waste must be stored securely to prevent exposure to radiation over long periods.

Due to radiation, nuclear waste undergoes a gradual reduction in its radioactivity, known as decay. Over time, this material becomes less harmful, though this process can span thousands of years. Finding a safe and long-term solution for nuclear waste storage is crucial to protect and preserve our environment.

• Most nuclear waste remains hazardous for many generations, necessitating intricate and secure storage solutions.
• Nuclear waste can either be high-level or low-level, defined by the waste's radioactivity and origins.

Proper management of nuclear waste is vital for safety and environmental health.

Decay Rate

Decay rate, or the rate at which a substance loses its form or effectiveness over time, is a crucial concept in understanding nuclear waste deactivation. Specifically, for radioactive materials, the decay rate indicates how quickly radioactive nuclei lose their activity, becoming stable non-radioactive matter.

The decay process is measured by half-life, which is the time it takes for half of the radioactive substance to decay. This parameter is unique for each radioactive material. In the case presented, a 3% decay rate per century represents how quickly the nuclear waste reduces its harmfulness over time.

• A consistent decay rate allows us to predict how much of the waste will remain over long periods.
• The decay rate is generally expressed as a percentage.
• This percentage helps to calculate the remaining amount using mathematical formulas.

This decay measurement is essential in formulating strategies and policies for waste storage and reducing environmental hazards.

Exponential Functions

Exponential functions are mathematical expressions that model situations where growth or decay happens at a continuously proportional rate. These functions are especially vital in calculating the decay of substances like nuclear waste over time.

The standard form of an exponential function is given as \( y = a(b)^x \), where:

• \( y \) is the final amount.
• \( a \) represents the initial amount.
• \( b \) is the growth (or decay) factor.
• \( x \) signifies the time.

For decay, as in the case of nuclear waste, the base \( b \) is less than 1, representing a decay factor. The exercise used this principle with the formula \( y = 150(0.97)^{10} \).

Through exponential functions, we can predict future scenarios based on current data, such as determining how much waste remains after a specified time. By substituting values into the function, various future scenarios and outcomes can be calculated quickly and accurately. Understanding these functions aids in planning and decision-making in scientific, business, and financial contexts.