Problem 23 - Applied Calculus

Question

Search and rescue teams work to find lost hikers. Members of the search team separate and walk parallel to one another through the area to be searched. Table 1.7 shows the percent, \(P\), of lost individuals found for various separation distances, \(d\), of the searchers. 18 $$\begin{array}{l|ccccc}\hline \text { Scparation distance } d \text { (ft) } & 20 & 40 & 60 & 80 & 100 \\\\\hline \text { Approximate percent found. } P & 90 & 80 & 70 & 60 & 50 \\\\\hline\end{array}$$ (a) Explain how you know that the percent found, \(P\), could be a linear function of separation distance, \(d\). (b) Find \(P\) as a linear function of \(d\). (c) What is the slope of the function? Give units and interpret the answer. (d) What are the vertical and horizontal intercepts of the function? Give units and interpret the answers.

Step-by-Step Solution

Answer

(a) The consistent decrease indicates linearity. (b) \( P = -0.5d + 100 \). (c) Slope: -0.5%/ft, meaning a 0.5% decrease per foot. (d) Intercepts: 100% at 0 ft and 0% at 200 ft; meaningfully they state detection capacity at those distances.

Step 1: Understanding Linear Function

We begin by understanding that a linear function can be represented in the form \( P = m \cdot d + b \), where \( m \) is the slope and \( b \) is the y-intercept. Given that the changes in \( P \) for equal intervals of \( d \) are consistent (i.e., the percent found decreases by the same amount as \( d \) increases by 20 ft), this suggests a linear relationship.

Step 2: Finding the Slope

To find the slope \( m \), calculate the change in \( P \) divided by the change in \( d \). Between 20 ft to 40 ft, \( P \) changes from 90 to 80, so the slope is \( m = \frac{80 - 90}{40 - 20} = \frac{-10}{20} = -0.5 \). This slope is consistent across other intervals.

Step 3: Determining the Linear Function

Using the slope \( m = -0.5 \) and a point from the table (e.g., when \( d = 20 \, ft \), \( P = 90 \)), substitute into the line equation: \( 90 = -0.5 \cdot 20 + b \). Solving for \( b \), we have \( 90 = -10 + b \) which gives \( b = 100 \). Therefore, the function is \( P = -0.5d + 100 \).

Step 4: Calculating the Slope Units and Interpretation

The slope \( -0.5 \) means that for each increase of 1 ft in separation distance \( d \), the percentage of individuals found decreases by 0.5%. The units are percent per foot.

Step 5: Identifying Vertical and Horizontal Intercepts

The vertical intercept occurs when \( d = 0 \), which gives \( P = 100 \). This suggests that if searchers have no separation, theoretically, they could find 100% of lost individuals. The horizontal intercept is when \( P = 0 \), found by solving \( 0 = -0.5d + 100 \), giving \( d = 200 \, ft \). This implies that if the separation is 200 ft, no individuals would be found.

Key Concepts

Slope and Intercept

A linear function is characterized by a constant slope and a y-intercept, forming a straight line on a graph. The slope is a measure of how steep the line is, indicating how much the dependent variable (often y) changes with a unit change in the independent variable (x). In the context of our exercise, the percent of lost individuals found decreases linearly as the searchers' separation, denoted by distance \(d\), increases.

The slope \(-0.5\) in our problem describes how quickly the percent decreases. Specifically, a negative slope of \(-0.5\) implies that for every 1 foot increase in separation, the percent found, \(P\), decreases by 0.5%. This provides a direct, simple interpretation of the rate of change of the function.

The y-intercept, here given as \(100\) when \(d = 0\), indicates the theoretical starting value when the independent variable is zero. It suggests that if there were no separation at all, 100% of individuals would be found. Together, the slope and y-intercept uniquely define the linear relationship in this search-and-rescue scenario.

Linear Equation

Linear equations model relationships exhibiting constant rates of change and can be expressed in the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. In this context, the equation \(P = -0.5d + 100\) represents the linear relationship between searchers' separation distance \(d\) and the percent of lost individuals found \(P\).

This equation was derived by calculating the slope \(m\), as the consistent change in \(P\) per increment in \(d\). The linear model makes predicting outcomes straightforward - for example, by entering a separation distance into this formula, you can easily estimate the percentage of individuals likely to be found. It simplifies complex real-world data into manageable insights.

Understanding this equation helps solve problems similar to ours by following a predictable pattern. A linear equation can help in visualizing and interpreting data trends, making complex decision-making processes more accessible, especially when variables are numerous or unpredictable.

Function Interpretation

Interpreting a linear function involves understanding what each component represents in real-world situations. For our problem, the linear function \(P = -0.5d + 100\) provides key insights into the relationship between searcher separation and search effectiveness.

The function shows us that as separation increases, the ability to find lost individuals decreases, quantified by the slope \(-0.5\). Practical decisions, such as finding the optimal spacing for the searchers, can be derived from such interpretations because it directly impacts efficiency and safety.

The intercepts offer meaningful starting and ending points for the function: the y-intercept at \(P = 100\%, d = 0\) represents the maximum potential success rate if no separation occurs, while the x-intercept at \(d = 200\, ft\) predicts a total failure to locate anyone. Each point highlights essential information that aids in strategic planning, ensuring resource allocation is efficient and effective in search and rescue missions. Understanding these interpretations is crucial for practical applications, encouraging informed decision-making based on quantitative analysis.