Problem 9 - Precalculus with Calculus Previews

Question

Initially 200 milligrams of a radioactive substance was present. After 6 hours the mass had decreased by \(3 \%\). Construct an exponential model \(A(t)=A_{0} e^{k t}\) for the amount remaining of the decaying substance after \(t\) hours. Find the amount remaining after 24 hours.

Step-by-Step Solution

Answer

After 24 hours, approximately 188.49 mg of the substance remains.

Step 1: Identify Given Values

Initially, the mass of the substance is 200 mg. After 6 hours, the mass decreases by 3%. This means that after 6 hours, the remaining mass is 97% of the initial mass: \[ A(6) = 0.97 \times 200 = 194 \text{ mg}.\]

Step 2: Set Up the Exponential Decay Model

The exponential decay model is given by \[ A(t) = A_0 e^{kt}, \] where \(A_0\) is the initial amount (200 mg) and \(k\) is the decay constant. We need to find \(k\) using the information that after 6 hours, \(A(6) = 194\).

Step 3: Solve for the Decay Constant \(k\)

Plug the known values into the equation:\[ 194 = 200 e^{6k}. \] Divide both sides by 200:\[ \frac{194}{200} = e^{6k}. \] Take the natural logarithm of both sides:\[ \ln\left(\frac{194}{200}\right) = 6k. \] Solve for \(k\): \[ k = \frac{\ln(0.97)}{6}. \]

Step 4: Substitute \(k\) Back into the Model

Substitute the value of \(k\) into the model to get:\[ A(t) = 200 e^{\left(\frac{\ln(0.97)}{6}\right)t}. \]

Step 5: Calculate the Amount After 24 Hours

Use the model to find the amount remaining after 24 hours:\[ A(24) = 200 e^{\left(\frac{\ln(0.97)}{6}\right)24}. \] Simplify the expression to calculate \(A(24)\).

Step 6: Compute the Simplified Expression

Simplify the computation:\[ A(24) = 200 \times (0.97)^{4} \approx 188.49 \text{ mg}. \]

Key Concepts

Radioactive Substance

Radioactive substances are materials that contain unstable atoms. These atoms undergo a process known as radioactive decay, in which they lose energy by emitting radiation. This transformation occurs spontaneously over time. For example, uranium and plutonium are well-known radioactive substances. When we deal with a radioactive substance, its quantity diminishes over time. This is crucial when considering exposure levels and safety when handling these materials in practical situations. Radioactivity is measured based on its half-life, which is the time needed for half of the radioactive atoms to decay. However, our problem specifically deals with a percentage decay over a set period, giving us a practical scenario related to real-life applications.

Decay Constant

In the context of radioactive decay, the decay constant, denoted as \(k\), is an essential component that describes the rate at which the substance decays. The decay constant is associated with the exponential decay model equation \(A(t) = A_0 e^{kt}\), where \(A(t)\) is the quantity at time \(t\), and \(A_0\) is the initial amount.The decay constant is specific to each substance:

• A larger \(k\) means the substance decays rapidly.
• A smaller \(k\) indicates slower decay over time.

To find \(k\) in our exercise, we use the known quantities of the substance's decay over a given time frame. As calculated, \(k = \frac{\ln(0.97)}{6}\) shows us how to find the decay constant using the natural logarithm based on the substance's percentage remained over 6 hours.

Exponential Model

The exponential model provides a mathematical framework for modeling the decay of a radioactive substance. The general form of the model is \(A(t) = A_0 e^{kt}\), where:

• \(A(t)\) is the amount of substance that remains after time \(t\).
• \(A_0\) is the initial amount of the substance, in our case, 200 mg.
• \(e\) is Euler's number, approximately 2.71828, indicating a natural exponential growth or decay.
• \(k\) is the decay constant.

This model helps us understand how quickly a substance decays over time. In the provided example, plugging the value of \(k\) into the model lets us predict the amount of substance remaining after any given time, such as 24 hours, ensuring accurate predictions of the decaying substance.

Natural Logarithm

The natural logarithm is a fundamental mathematical function with the base \(e\), represented as \(\ln(x)\). It is integral in solving exponential equations, especially when dealing with continuous growth or decay scenarios like radioactive decay.In our problem, we employ the natural logarithm to solve for the decay constant \(k\):

• It helps us convert the exponential part of the equation into a linear form, which is easier to handle mathematically.
• It allows us to isolate \(k\) by taking the natural logarithm of both sides of the equation \(e^{6k} = \frac{194}{200}\).

Using \(\ln\) is a standard technique for resolving equations of this nature because it transforms multiplicative processes into additive ones, simplifying the pathway to finding unknowns like the decay constant. Overall, the natural logarithm is a powerful tool in modeling and understanding exponential processes.