Problem 25 - Calculus: Graphical, Numerical, Algebraic

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 amount of radon in a sealed sample of air to decay to 90\(\%\) of its original value?

Step-by-Step Solution

Answer

It will take approximately 6 days for the amount of radon in a sealed sample of air to decay to 90% of its original value.

Step 1: Understand the Decay Equation

The decay equation given is \(y = y_{0} e^{-0.18 t}\), where \(y\) is the amount of radon remaining, \(y_{0}\) is the initial amount of radon, \(e\) is Euler's number (approximately equal to 2.71828), and \(t\) is time in days.

Step 2: Formulate the Decay Equation

We are looking for the time \(t\) it takes for the radon to decay to 90% of the original value. So, we set \(y\) to 0.9\(y_{0}\), which results the in the equation 0.9\(y_{0} = y_{0} e^{-0.18 t}\).

Step 3: Simplify the Equation

The \(y_{0}\) on each side cancels out, which leaves 0.9 = \(e^{-0.18 t}\). Then, we take the natural logarithm (ln) of both sides to remove the exponent on \(e\), getting \(ln(0.9) = -0.18 t\).

Step 4: Solve for Time t

Solving the equation \(ln(0.9) = -0.18 t\) for \(t\) gives us \(t = \frac{ln(0.9)}{-0.18}\).

Step 5: Calculate Time

Finally, calculating the value of \(t = \frac{ln(0.9)}{-0.18}\) gives us approximately 5.99 days.

Key Concepts

Exponential Decay

Exponential decay describes the process by which a quantity diminishes at a rate proportional to its current value. This phenomenon is frequently observed in nature and technology, such as the decay of radioactive substances, fading of light, or depreciation of assets over time.

The mathematical representation of exponential decay is typically formulated as \( y = y_0 e^{-kt} \), where:\

• \( y \) represents the remaining amount of the substance at time \( t \).
• \( y_0 \) is the initial amount of the substance.
• \( e \) is Euler's number, a fundamental constant in mathematics.
• \( k \) is the decay constant that signifies the decay's speed.
• \( t \) is the time that has passed since the decay began.

Understanding the mechanics of exponential decay aids vastly in various scientific and financial calculations. In the context of radon decay, knowing the decay constant allows for predicting how rapidly the substance diminishes over time.

Euler's Number

Euler's number, denoted as \( e \), is an irrational and transcendental number approximately equal to 2.71828. It is one of the most important constants in mathematics and appears frequently in problems of growth and decay, particularly when these changes happen continuously over time.

Euler's number arises in the study of compound interest, population growth, and in calculus as the base of the natural logarithm. It is the limit of \( (1 + 1/n)^n \) as \( n \) approaches infinity, which can be thought of as reflecting continuous compounding to infinity.

Importance in Exponential Functions

In our radon decay example, the presence of \( e \) in the decay equation emphasizes that the change is happening exponentially. Therefore, \( e \) serves as the base of the exponential function, symbolizing the continuous and natural change rate of the decaying substance.

Natural Logarithm

The natural logarithm is the logarithm to the base of Euler's number \( e \). It is often denoted as \( ln \) and is a critical tool for solving exponential equations where the unknown variable is in the exponent place.

For the equation \( e^x = b \), taking the natural logarithm of both sides yields \( ln(e^x) = ln(b) \), which simplifies to \( x = ln(b) \) because the natural logarithm and \( e \) are inverse operations. This property makes it possible to isolate the variable \( x \) and solve for it when dealing with exponential growth or decay.

Using Natural Logarithm to Solve Decay Problems

In the case of radon-222 decay, to find how long it takes for a sample to decay to 90% of its original value, we employ the natural logarithm to solve for the time variable \( t \). By applying \( ln \) to both sides of our proportionate decay equation, we can unravel the exponent and compute the time needed for the decay process. Understanding the concept of natural logarithms is pivotal for demystifying the calculations involved in exponential decay scenarios.