Problem 21 - Precalculus Mathematics for Calculus

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) 13 kg (b) Approximately 6.63 kg

Step 1: Understand the Given Function

The function given is \( m(t) = 13e^{-0.015t} \) which describes the mass of a radioactive substance remaining after \( t \) days. Here, 13 represents the initial mass (when \( t = 0 \)), and the exponent \(-0.015t\) defines the rate of decay over time.

Step 2: Calculate Mass at Time \( t = 0 \)

To find the mass at time \( t = 0 \), substitute \( t = 0 \) into the function: \( m(0) = 13e^{-0.015\times0} = 13e^0 \). Since \( e^0 = 1 \), we have \( m(0) = 13 \times 1 = 13 \text{ kg} \).

Step 3: Substitute 45 Days into the Function

To find out how much mass remains after 45 days, replace \( t \) with 45 in the function: \( m(45) = 13e^{-0.015\times45} \).

Step 4: Calculate the Exponential Part

Compute the exponent: \(-0.015 \times 45 = -0.675\). Now substitute it back: \( m(45) = 13e^{-0.675} \).

Step 5: Evaluate the Exponential Function

Using a calculator, find \( e^{-0.675} \) which is approximately 0.5097. Substitute back to get \( m(45) = 13 \times 0.5097 \).

Step 6: Final Calculation for Remaining Mass

Multiply the two values: \( 13 \times 0.5097 \approx 6.6261 \text{ kg} \). Thus, after 45 days, approximately 6.6261 kg of mass remains.

Key Concepts

Exponential Function

An exponential function describes a process where the rate of change is proportional to the quantity present. It is typically characterized by the general form \( f(t) = a e^{bt} \), where \( a \) is the initial amount, \( e \) is the base of the natural logarithms, and \( b \) is the rate of growth or decay. In the context of radioactive decay, the function becomes a decrease over time.
For our given problem, the function is \( m(t) = 13e^{-0.015t} \). This tells us how the mass of a radioactive material decreases as time progresses.

• \( e \) is approximately 2.718, which is a constant.
• The term \(-0.015t\) is in the exponent, indicating an exponential decay.

Exponential decay means that the quantity diminishes at a rate proportional to its current value, leading to a characteristic rapid drop that slows over time as the remaining amount decreases.Understanding exponential functions is crucial as they describe various natural phenomena, including population growth, bank interest, and, as in this case, radioactive decay.

Mass Function

The mass function is an application of the exponential function used to describe how the mass of a substance changes over time. The given mass function \( m(t) = 13e^{-0.015t} \) models the decay of a radioactive substance.
This function provides us with the mass \( m(t) \) at any given time \( t \). Let's break it down piece by piece:

• \( 13 \) kg represents the initial mass of the radioactive substance when \( t = 0 \). This is the starting point before any decay has occurred.
• The exponential term \( e^{-0.015t} \) tells us how the mass changes as time \( t \) increases.

The function can be used to make predictions about the remaining mass at any given time, which is crucial in fields like nuclear physics and environmental science, where understanding radioactive decay is essential for both safety and practical applications.

Decay Rate

The decay rate in radioactive decay is a measure of how quickly the substance is losing mass over time. It plays a pivotal role in the exponential function utilized to model decay. In our function \( m(t) = 13e^{-0.015t} \), the decay rate is represented by the exponent \(-0.015\).
Here’s what that number tells us:

• The negative sign indicates that the process is a decay—a decrease over time.
• \( 0.015 \) is the constant that quantifies how rapidly the substance is decaying.

This decay rate translates to a gradual reduction in mass, causing the curve to slope downward. It allows us to compute exactly how much mass remains after any given period simply by plugging in \( t \) into the function.
Understanding the decay rate is fundamental to predicting how long a radioactive material will remain potent or hazardous. It helps in calculating the half-life, the time required for half of the material to decay, which is a critical component in fields like nuclear medicine and archaeological dating.