Problem 89 - A Graphical Approach to College Algebra

Question

Distance to Lightning When a bolt of lightning strikes in the distance, there is often a delay between seeing the lightning and hearing the thunder. The function \(f(x)=\frac{x}{5}\) computes the approximate distance in miles between an observer and a bolt of lightning when the delay is \(x\) seconds. (a) Find \(f(15)\) and interpret the result. (b) Graph \(y=f(x) .\) Let the domain of \(f\) be \([0,20]\)

Step-by-Step Solution

Answer

(a) 3 miles; (b) Line from (0,0) to (20,4).

Step 1: Understand the Problem

This problem revolves around the function \( f(x) = \frac{x}{5} \), which calculates the distance in miles based on the delay \( x \) in seconds between seeing lightning and hearing thunder. We need to find the distance for a 15-second delay and graph the function for a domain from 0 to 20.

Step 2: Calculate \( f(15) \)

To find \( f(15) \), substitute \( x = 15 \) into the function: \( f(15) = \frac{15}{5} = 3 \). This means that with a 15-second delay, the lightning is approximately 3 miles away.

Step 3: Interpret the Result

The result \( f(15) = 3 \) indicates that during a 15-second delay between seeing lightning and hearing thunder, the bolt of lightning is approximately 3 miles away from the observer.

Step 4: Set Up the Graph

Prepare to graph the function \( y = f(x) = \frac{x}{5} \) over the domain \([0,20]\). This implies the graph will be a line starting from (0,0) and moving to (20,4).

Step 5: Plot Key Points

Calculate key points for clarity in the graph. When \( x = 0 \), \( y = f(0) = 0 \). When \( x = 20 \), \( y = f(20) = 4 \). These endpoints will form a straight line.

Step 6: Draw the Graph Line

Draw a straight line connecting these calculated points (0,0) and (20,4) on the coordinate plane. This linear graph represents the function \( f(x) = \frac{x}{5} \).

Key Concepts

Distance Calculation

The concept of calculating distance using a function involves understanding how different variables interact with each other. In this problem, the delay between seeing lightning and hearing thunder is used to estimate the distance the lightning is from an observer. The function given is:

• \( f(x) = \frac{x}{5} \)

This equation provides a quick estimation of distance in miles by dividing the number of elapsed seconds by 5.
Suppose the delay, \( x \), is 15 seconds. By substituting \( x = 15 \) into the function, we calculate:

• \( f(15) = \frac{15}{5} = 3 \)

Thus, a 15-second delay indicates that the lightning strike is approximately 3 miles away from the observer.

Linear Function

A linear function represents a straight line on a graph, characterized by a constant rate of change or slope. In the context of this problem, the function \( f(x) = \frac{x}{5} \) is linear.
This means that for every increase of 1 second in the delay \( x \), the distance increases by 0.2 miles.

• This slope of \( \frac{1}{5} \) indicates that the graph of the function will be a line with this constant upward trend.

The linearity implies predictability and simplicity in determining distances based solely on the time delay, making calculations straightforward and intuitive.

Domain and Range

The domain and range of a function provide insights into the possible input values and the corresponding output values the function can have.
For the function \( f(x) = \frac{x}{5} \), we're interested in a limited domain:

• Domain: \([0, 20]\)

This means we're only considering time delays from 0 to 20 seconds. Consequently, the range, or the possible distances, becomes:

• Range: \([0, 4]\)

This range indicates that distances will vary from 0 to 4 miles, which is directly determined by the maximum delay time considered in this problem. Understanding the domain and range helps accurately set up and interpret the graph.

Mathematical Interpretation

Mathematical interpretation involves understanding not just how to compute a result, but what that result means in real-world terms.
In this exercise, the output of the function represents a meaningful physical quantity—distance. When we calculated \( f(15) = 3 \), this mathematically correlates the 15-second delay to the approximate distance of 3 miles.

• This conceptualization is essential because it allows observers to appropriately assess their proximity to a lightning strike.
• Graphs further aid this interpretation by visually representing how changes in delay translate to changes in distance.

Thus, the function serves as a practical tool to convert observable phenomena (delay) into actionable insights (distance).