Problem 17 - Thomas Calculus

Question

Radon- 222 . The decay equation for radon-222 gas is known to be \(y=y_{0} e^{-0.18 t},\) with \(t\) in days. About how long will it take the radon in a sealed sample of air to fall to 90\(\%\) of its original value?

Step-by-Step Solution

Answer

It takes approximately 0.59 days for the radon to reduce to 90% of its original value.

Step 1: Identify Given Information

We are given the decay equation \( y = y_0 e^{-0.18t} \).We are asked to find the time \( t \) when the remaining radon is 90\% of the original amount \( y_0 \). In mathematical terms, we want \( y = 0.9y_0 \).

Step 2: Set Up the Equation

Substitute \( y = 0.9y_0 \) into the decay equation: \( 0.9y_0 = y_0 e^{-0.18t} \).

Step 3: Simplify the Equation

Divide both sides of the equation by \( y_0 \) to simplify:\( 0.9 = e^{-0.18t} \).

Step 4: Take the Natural Logarithm

Apply the natural logarithm to both sides to solve for \( t \):\( \ln(0.9) = \ln(e^{-0.18t}) \).This simplifies to \( \ln(0.9) = -0.18t \).

Step 5: Solve for Time \( t \)

Rearrange the equation to isolate \( t \):\( t = \frac{\ln(0.9)}{-0.18} \).Calculate \( t \):\( \ln(0.9) \approx -0.10536 \),\( t \approx \frac{-0.10536}{-0.18} \approx 0.585 \) days.

Key Concepts

Radon-222

Radon-222 is a naturally occurring radioactive gas resulting from the decay of uranium and thorium in the Earth's crust. It is a heavy gas that is odorless, colorless, and tasteless, making it difficult to detect without specialized equipment. Radon-222 is considered hazardous to health, particularly when it accumulates in indoor spaces such as homes and workplaces.

Some key points about Radon-222 include:

• It has a half-life of approximately 3.8 days, meaning half of the radon will decay over this period.
• As a radioactive gas, it emits alpha particles as it decays, which can damage lung tissue if inhaled.
• Long-term exposure to elevated radon levels can increase the risk of lung cancer.

Understanding the decay process of radon-222 helps in assessing risks and implementing safety measures to mitigate its health impacts. Monitoring radon levels and improving ventilation are common methods to reduce indoor radon concentrations.

Exponential Decay

Exponential decay refers to a process where quantities diminish at a rate proportional to their current value. This is a common pattern seen in natural decay processes, such as radioactive decay, which is characterized by a constant decay rate.

In the equation for radon-222 decay, given by \( y = y_0 e^{-0.18t} \), this form illustrates exponential decay. Here:

• \( y_0 \) represents the initial amount.
• \( y \) is the amount remaining at time \( t \).
• The decay constant, \( 0.18 \), dictates how quickly the quantity decreases.

As time advances, the exponential factor \( e^{-0.18t} \) reduces the original amount, \( y_0 \), resulting in a smaller and smaller remaining quantity. This mathematical model gives precise predictions on how radon levels decrease over time, which is crucial for understanding radon's behavior in sealed environments.

Natural Logarithm

The natural logarithm, often denoted as \( \ln \), is a logarithm with base \( e \), where \( e \) is approximately equal to 2.718. It plays an essential role in solving equations involving exponential functions, such as those occurring in decay processes.

In the exercise involving radon-222, the natural logarithm is used to isolate the variable \( t \) in the problem. The relevant step is:

• After simplifying the decay equation to \( 0.9 = e^{-0.18t} \), we take the natural logarithm of both sides: \( \ln(0.9) = -0.18t \). This transforms the equation from an exponential form to one that can be easily solved for \( t \).

By applying the properties of logarithms, particularly that \( \ln(e^x) = x \), one can solve complex exponential equations intuitively. Natural logarithms are crucial in mathematics for understanding and manipulating exponential relationships, aiding in solving a diverse range of real-world problems.