Problem 54 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

RADIOACTIVE DECAY The following data were compiled by a researcher during an experiment designed to study the decay of a radioactive substance: \begin{tabular}{l|c|c} Number of hours & 0 & 5 \\ \hline Grams of substance & 1,000 & 700 \end{tabular} Assuming that the sample of radioactive substance decays exponentially, how much is left after 20 hours?

Step-by-Step Solution

Answer

The amount remaining after 20 hours is approximately 238 grams.

Step 1 - Understanding Exponential Decay

In an exponential decay process, the amount of substance can be modeled with the equation: \[N(t) = N_0 e^{-\beta t}\] where: - \(N(t)\) is the amount of substance remaining after time \(t\). - \(N_0\) is the initial amount of the substance. - \(\beta\) is the decay constant. - \(t\) is the elapsed time.

Step 2 - Substituting Known Values

From the problem, it is known that at \(t = 0\) hours, the amount of substance \(N_0 = 1000\) grams. After \(t = 5\) hours, \(N(5) = 700\) grams. Substitute these values into the decay equation to find the decay constant \(\beta\): \[700 = 1000 e^{-5\beta}\]

Step 3 - Solving for the Decay Constant

To solve for \(\beta\), first isolate the exponential term: \[\frac{700}{1000} = e^{-5\beta}\] which simplifies to: \[0.7 = e^{-5\beta}\] Take the natural logarithm of both sides to solve for \(\beta\): \[\text{ln}(0.7) = -5\beta\] \[\beta = -\frac{\text{ln}(0.7)}{5}\]

Step 4 - Calculating the Decay Constant

Calculate \(\beta\) using the natural logarithm: \[\beta = -\frac{\text{ln}(0.7)}{5} \ \beta \approx 0.0718 \text{ per hour}\]

Step 5 - Applying the Decay Equation

Now, use the decay equation with \(\beta = 0.0718\) to find the amount of substance remaining after \(t = 20\) hours: \[N(20) = 1000 e^{-0.0718 \times 20}\]

Step 6 - Calculating the Amount Remaining

Calculate the amount remaining: \[N(20) = 1000 e^{-1.436}\] \[N(20) \approx 1000 \times 0.238\] \[N(20) \approx 238 \text{ grams}\]

Key Concepts

Exponential Decay

In exponential decay, a quantity decreases at a rate proportional to its current value. This concept is common in natural processes like radioactive decay. The decay can be described using the formula: \[N(t) = N_0 e^{-\beta t}\] Here:

• \(N(t)\) is the remaining amount of substance at time \(t\).
• \(N_0\) is the initial amount.
• \(\beta\) is the decay constant, reflecting how quickly the substance decays.
• \(e\) is the base of the natural logarithms.

The equation captures the essence of exponential decay, showing that as time increases, the amount of the substance decreases exponentially.

Decay Constant

The decay constant \(\beta\) is a critical component in understanding radioactive decay. It determines the speed at which a substance decays. You can find \(\beta\) using the known quantities at two different times. For example, if you know the amount of substance at times \(t_0\) and \(t_1\), you can set up the equation:
\[\frac{N(t_1)}{N_0} = e^{-\beta t_1}\]
In our example, after 5 hours, 700 grams remain from an initial 1000 grams. Substitute these into the equation to get:
\[\frac{700}{1000} = e^{-5\beta}\]
Taking the natural logarithm on both sides helps isolate \(\beta\):
\[\text{ln}(0.7) = -5\beta\]
Solving for \(\beta\), you get:
\[\beta = -\frac{\text{ln}(0.7)}{5} \approx 0.0718 \text{ per hour}\]

Natural Logarithm

The natural logarithm (\text{ln}) is the logarithm to the base \(e\), where \(e\) is approximately 2.718. It's widely used in growth and decay models because it simplifies the equations.
In our exercise, we use the natural logarithm to isolate and solve for the decay constant \(\beta\). For example:
\[\text{ln}(0.7) = -5\beta\]
Natural logarithms make solving exponential equations straightforward, as they can linearize the exponential terms. It's a powerful tool for many scientific calculations, particularly those involving growth and decay.

Half-life

Half-life is the time required for a quantity to reduce to half its initial value. It's another important concept in radioactive decay. The half-life \(t_{1/2}\) is related to the decay constant \(\beta\) by the formula:
\[t_{1/2} = \frac{\text{ln}(2)}{\beta}\] For example, if \(\beta = 0.0718 \text { per hour}\), the half-life is:
\[t_{1/2} = \frac{\text{ln}(2)}{0.0718} \approx 9.65 \text{ hours}\]
Understanding half-life helps in many fields, from predicting how long a radioactive substance will remain hazardous to calculating the time required for medications to lose effectiveness in the body.