Question

In Exercises, use a graphing utility to graph the function. Be sure to choose an appropriate viewing window. $$ N(t)=500 e^{-0.2 t} $$

Short answer

Graph the function \(N(t)=500 e^{-0.2 t}\) using a graphing utility with a suitable viewing window, understanding that as \(t\) increases, \(N(t)\) decreases gradually toward zero, typical of decay processes.

Step-by-step solution

Understanding the Function

This is an exponential decay function since it has a negative exponent. It models processes such as radioactive decay or population growth in a constrained environment. The number 500 acts as the initial value (when time \(t\) equals zero). Consequently, as \(t\) increases, \(N(t)\) will decrease.

Setting up the Graphing Utility

To correctly display the function, a suitable viewing window needs to be set. Since it's not specified, a default viewing window can be used, for example for \(t\) ranging from 0 to 10 and \(N(t)\) ranging from 0 to 600.

Graphing the Function

Enter the function \(N(t)=500 e^{-0.2 t}\) into the graphing utility and plot the graph. The function should start at 500 when \(t=0\) and then it will decay over time, gradually getting closer to the \(t\)-axis.

Analysis of the Graph

Analyze the graph. As expected, the graph should show the exponential decay, starting at 500 and getting closer to the horizontal axis as time increases. This property is typical of decay processes modeled by such a function

Key concepts

Radioactive Decay Basics

Radioactive decay is a natural process where substances lose energy over time by emitting radiation. It's a practical example of an exponential decay function, which is crucial in many scientific fields, such as physics and chemistry. Radioactive elements have unstable nuclei that spontaneously emit particles, leading to a decrease in the number of these unstable atoms with time. This decay happens at a predictable rate, usually measured in terms of a half-life—the time it takes for half of a sample to decay.

• The decay is characterized by a continuous decrease modeled mathematically using exponential functions.
• An essential term in radioactive decay is the 'decay constant,' which directly influences how quickly the substance decays over time.
• The equation for radioactive decay can be written as: \[N(t) = N_0 e^{-kt}\]where \( N_0 \) is the initial quantity of substance, \( k \) is the decay constant, and \( t \) is time.

Understanding these concepts helps scientists predict how long a radioactive substance will persist, which is very important for applications such as nuclear medicine and radiocarbon dating.

Using a Graphing Utility Effectively

Graphing utilities are powerful tools used to visualize mathematical functions. They help us better understand complex behaviors that are difficult to observe through equations alone. Here’s how to use a graphing utility to explore the function \(N(t) = 500 e^{-0.2 t}\):

• First, input the function into the graphing utility. Many calculators and software applications allow for this.
• Then, set an appropriate viewing window. For exponential decay, it's helpful to choose a window that displays the most interesting part of the graph, typically where the graph starts at the initial value and shows the decay over time.
• A common setting for exponential decay might be setting \(t\) from 0 to 10 and \(N(t)\) from 0 to slightly above the initial value, such as 600, ensuring the initial part of the decay is visible.

By effectively setting up your graphing utility, you can more easily observe the specific characteristics of exponential decay functions, such as how they approach the \(t\)-axis without actually touching it.

Understanding Exponential Functions

Exponential functions are a type of mathematical function characterized by their constant rate of growth or decay. These functions are in the form \( y = a e^{bx} \), where \( a \) is the initial value, \( e \) is Euler's number (approximately 2.71828), \( b \) is the rate of growth or decay, and \( x \) is the variable. The specific form of the function determines whether it models growth or decay:

• When \( b \) is positive, the function represents exponential growth, exemplifying processes such as population growth and compound interest.
• When \( b \) is negative, as in \(N(t) = 500 e^{-0.2 t}\), the function models exponential decay, illustrating phenomena like radioactive decay and depreciation of assets.
• Exponential functions have unique properties, such as never reaching zero. They asymptotically approach the \(x\)-axis (or \(t\)-axis in our case), meaning they get closer indefinitely without touching it.

Understanding exponential functions is crucial, as they are prevalent in various scientific, economic, and everyday contexts, often serving as models for real-world phenomena.