Problem 65 - Algebra and Trigonometry

Question

Radioactive Decay A radioactive substance decays in such a way that the amount of mass remaining after \(t\) days is given by the function $$ m(t)=13 e^{-0.015 t} $$ where \(m(t)\) is measured in kilograms. (a) Find the mass at time \(t=0\) . (b) How much of the mass remains after 45 days?

Step-by-Step Solution

Answer

(a) The mass at time \(t=0\) is 13 kg. (b) After 45 days, approximately 6.62 kg remains.

Step 1: Understanding the Decay Function

The decay function is given as \( m(t) = 13 e^{-0.015 t} \). Here, \( m(t) \) is the mass remaining at time \( t \), and 13 kilograms is the initial mass of the substance.

Step 2: Finding Mass at Time t = 0

To find the mass at time \( t = 0 \), substitute \( t = 0 \) into the function: \[m(0) = 13 e^{-0.015 \times 0} = 13 e^0 = 13 \times 1 = 13.\]Thus, the initial mass is 13 kilograms.

Step 3: Substituting Time t = 45

To find the mass after 45 days, substitute \( t = 45 \) into the decay function: \[m(45) = 13 e^{-0.015 \times 45}.\]Calculate \( -0.015 \times 45 = -0.675 \). So, we have \[\m(45) = 13 e^{-0.675}.\]

Step 4: Calculating the Exponent

Using a calculator, we calculate \( e^{-0.675} \). \[e^{-0.675} \approx 0.5096\]

Step 5: Finding Remaining Mass

Substitute the approximate value of \( e^{-0.675} \) into the expression for \( m(45) \):\[m(45) = 13 \times 0.5096 \approx 6.6248.\]Thus, the mass remaining after 45 days is approximately 6.62 kilograms.

Key Concepts

Exponential Decay

Exponential decay is a process that describes the reduction of a quantity at a consistent percentage rate over time. This concept, often represented by an exponential function, is crucial in understanding how certain processes evolve, such as radioactive decay. When a substance undergoes exponential decay, every fixed time period results in the same proportion of the remaining amount being lost. It's like watching dominoes fall, each one knocking the next over in a rhythmic sequence.
For example, in the radioactive decay problem, the mass of the substance decreases at an exponential rate defined by the function:

• The function is of the form: \( m(t) = m_0 e^{-kt} \), where \( m_0 \) is the initial quantity (13 kg for our case), \( e \) is the base of the natural logarithm, and \( k \) is the decay constant.
• This decay constant \( k \) influences how quickly the substance reduces. A larger \( k \) means a faster decay rate.
• In our example, the decay rate is 0.015 per day.

Exponential decay is essential to grasp because it applies to various fields, from physics to economics, explaining situations where quantities diminish over time.

Decay Function

The decay function is a mathematical expression that models the process by which quantities decrease over time. It's a kind of function specifically designed to handle exponential decay. Think of it as a tool in a scientist's toolkit for predicting how much of something will remain after a certain period.
In the provided exercise, the decay function is given by:

• The equation for the mass over time is \( m(t) = 13 e^{-0.015t} \).
• Here, \( m(t) \) represents the mass at time \( t \), and the initial mass is 13 kilograms.
• The term \( e^{-0.015t} \) shows how the mass decreases exponentially as time goes on.

Understanding this function is crucial because it reveals how quickly the mass of a radioactive element diminishes. By setting \( t = 0 \), we see the initial mass is unchanged (13 kg), and by solving for another time, like \( t = 45 \), we can calculate the quantity remaining. This example illustrates how decay functions are used to make predictions in real-life scenarios using math.

Exponential Function

The exponential function is a mathematical relation involving an exponent and a constant base known as \( e \), approximately equal to 2.718. This function is integral in handling many phenomena, such as growth and decay processes. In this scenario, our base \( e \) is raised to a power involving negative multiplication with the decay constant and time, which is why the decay is exponential.
In our context:

• The exponential function used is \( e^{-0.015t} \).
• It means that as time \( t \) increases, \( e^{-0.015t} \) decreases because the exponent is negative.
• This is what drives the decay process in the function \( m(t) = 13 e^{-0.015t} \).

Such functions are powerful because they efficiently describe many real-world processes, like cooling coffee or population decline, beyond radioactive decay. Their application helps resolve practical issues by providing precise computation over time, leading to accurate forecasts and decisions.