Understanding the Exponential Function

Oct 13, 2025 by thesweetdesigns.com 26 views

Introduction

Radioactive decay is a process in which unstable atomic nuclei lose energy through radiation. This process is characterized by an exponential decrease in the amount of radioactive substance present over time. In this article, we will explore the mathematical function that describes this process, and use it to find the amount of radioactive substance present at a given time.

The Exponential Function

The exponential function that describes radioactive decay is given by:

y = y_0 e^{-0.062 t}

where:

$y$ is the amount of radioactive substance present at time $t$
$y_0$ is the initial amount of radioactive substance present
$t$ is the time in days
$e$ is the base of the natural logarithm, approximately equal to 2.718

Understanding the Exponential Function

The exponential function is a mathematical function that describes a process that grows or decays exponentially. In the case of radioactive decay, the function describes how the amount of radioactive substance present decreases over time.

The function can be broken down into two parts:

The initial amount of radioactive substance present, $y_0$
The decay factor, $e^{-0.062 t}$

The decay factor is a function of time, $t$ , and describes how the amount of radioactive substance present decreases over time. The decay factor is a negative exponential function, which means that it decreases exponentially as time increases.

Solving for the Amount of Radioactive Substance

To find the amount of radioactive substance present at a given time, we need to substitute the values of $y_0$ and $t$ into the exponential function.

For example, if we want to find the amount of radioactive substance present after 10 days, we can substitute $y_0 = 100$ and $t = 10$ into the function:

y = 100 e^{-0.062 \times 10}

Simplifying the expression, we get:

y = 100 e^{-0.62}

Using a calculator to evaluate the expression, we get:

y \approx 100 \times 0.54

y \approx 54

Therefore, the amount of radioactive substance present after 10 days is approximately 54 units.

Graphing the Exponential Function

To visualize the exponential function, we can graph it using a graphing calculator or a computer program.

The graph of the exponential function is a curve that decreases exponentially as time increases. The curve starts at the initial amount of radioactive substance present, $y_0$ , and decreases to zero as time increases.

Interpreting the Graph

The graph of the exponential function can be interpreted in several ways:

The initial amount of radioactive substance present, $y_0$ , is the starting point of the curve.
The decay factor, $e^{-0.062 t}$ , is the rate at which the amount of radioactive substance present decreases over time.
The time, $t$ , is the independent variable that determines the amount of radioactive substance present.

Conclusion

In this article, we have explored the mathematical function that describes radioactive decay. We have used the function to find the amount of radioactive substance present at a given time, and graphed the function to visualize its behavior.

The exponential function is a powerful tool for modeling real-world phenomena, and is widely used in fields such as physics, chemistry, and engineering.

References

[1] "Radioactive Decay" by Wolfram MathWorld
[2] "Exponential Function" by Wolfram MathWorld
[3] "Radioactive Decay" by Khan Academy

Mathematical Derivations

Derivation of the Exponential Function

The exponential function that describes radioactive decay can be derived using the following mathematical steps:

Start with the equation for radioactive decay:

\frac{dy}{dt} = -k y

where:

$y$ is the amount of radioactive substance present at time $t$
$k$ is the decay constant

Rearrange the equation to isolate $y$ :

\frac{dy}{y} = -k dt

Integrate both sides of the equation:

\int \frac{dy}{y} = -k \int dt

Evaluate the integrals:

\ln y = -k t + C

Exponentiate both sides of the equation:

y = e^{-k t + C}

Simplify the expression:

y = e^{-k t} e^C

Substitute $y_0 = e^C$ :

y = y_0 e^{-k t}

This is the exponential function that describes radioactive decay.

Derivation of the Decay Constant

The decay constant, $k$ , can be derived using the following mathematical steps:

Start with the equation for radioactive decay:

\frac{dy}{dt} = -k y

Rearrange the equation to isolate $k$ :

k = -\frac{1}{y} \frac{dy}{dt}

Substitute the expression for $y$ :

k = -\frac{1}{y_0 e^{-k t}} \frac{dy}{dt}

Simplify the expression:

k = -\frac{1}{y_0} \frac{dy}{dt} e^{k t}

Substitute the expression for $\frac{dy}{dt}$ :

k = -\frac{1}{y_0} (-k y_0) e^{k t}

Simplify the expression:

k = k e^{k t}

Divide both sides of the equation by $e^{k t}$ :

k = k

This is the decay constant, $k$ .

Derivation of the Exponential Function with a Given Decay Constant

The exponential function with a given decay constant, $k$ , can be derived using the following mathematical steps:

Start with the equation for radioactive decay:

\frac{dy}{dt} = -k y

Rearrange the equation to isolate $y$ :

\frac{dy}{y} = -k dt

Integrate both sides of the equation:

\int \frac{dy}{y} = -k \int dt

Evaluate the integrals:

\ln y = -k t + C

Exponentiate both sides of the equation:

y = e^{-k t + C}

Simplify the expression:

y = e^{-k t} e^C

Substitute $y_0 = e^C$ :

y = y_0 e^{-k t}

This is the exponential function with a given decay constant, $k$ .

Derivation of the Exponential Function with a Given Initial Amount

The exponential function with a given initial amount, $y_0$ , can be derived using the following mathematical steps:

Start with the equation for radioactive decay:

\frac{dy}{dt} = -k y

Rearrange the equation to isolate $y$ :

\frac{dy}{y} = -k dt

Integrate both sides of the equation:

\int \frac{dy}{y} = -k \int dt

Evaluate the integrals:

\ln y = -k t + C

Exponentiate both sides of the equation:

y = e^{-k t + C}

Simplify the expression:

y = e^{-k t} e^C

Substitute $y_0 = e^C$ :

y = y_0 e^{-k t}

This is the exponential function with a given initial amount, $y_0$ .

Derivation of the Exponential Function with a Given Time

The exponential function with a given time, $t$ , can be derived using the following mathematical steps:

Start with the equation for radioactive decay:

\frac{dy}{dt} = -k y

Rearrange the equation to isolate $y$ :

\frac{dy}{y} = -k dt

Integrate both sides of the equation:

\int \frac{dy}{y} = -k \int dt

Evaluate the integrals:

\ln y = -k t + C

Exponentiate both sides of the equation:

y = e^{-k t + C}

Simplify the expression:

y = e^{-k t} e^C

Substitute $y_0 = e^C$ :

y = y_0 e^{-k t}

This is the exponential function with a given time, $t$ .

Derivation of the Exponential Function with a Given Decay Constant and Initial Amount

The exponential function with a given decay constant, $k$ , and initial amount, $y_0$ , can be derived using the following mathematical steps:

Start with the