Problem 23
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 \(1.5\) shows the percent, \(P\), of lost individuals found for various separation distances, \(d\), of the searchers. 16 $$ \begin{array}{l|lllll} \hline \text { Separation distance } d \text { (ft) } & 20 & 40 & 60 & 80 & 100 \\ \hline \text { Approximate percent found, } P & 90 & 80 & 70 & 60 & 50 \\ \hline \end{array} $$ (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
Verified Answer
(a) Constant rate change suggests linearity. (b) \(P = -0.5d + 100\). (c) Slope is -0.5%/ft; each 1ft increases, finds 0.5% less. (d) Vertical intercept: 100% at 0 ft; Horizontal: 200 ft, 0% found.
1Step 1: Identify Linear Relationship
Multiple separation distances and corresponding percentages are given. To determine if the relationship is linear, calculate the difference in percentages for equal differences in separation distances. Here, the difference in separation distance is consistently 20 ft, and the percent found decreases by 10 each time: 90 to 80, 80 to 70, etc. This constant rate of change suggests that the relationship between percent found, \(P\), and separation distance, \(d\), may be linear.
2Step 2: Calculate Slope
The rate of change (slope) can be calculated as the change in \(P\) divided by the change in \(d\). From the data, a change in \(d\) of 20 ft corresponds to a decrease in \(P\) of 10 percent. The slope \(m\) is given by \(m = \frac{\Delta P}{\Delta d} = \frac{-10}{20} = -0.5\).
3Step 3: Formulate Linear Equation
Using the slope of -0.5 and a point from the table, say \((20, 90)\), we can use the point-slope form: \(P - 90 = -0.5(d - 20)\). Simplifying, \(P = -0.5d + 100\), provides us with the linear equation relating \(P\) and \(d\).
4Step 4: Slope Interpretation
The slope of -0.5 implies that for every increase of 1 foot in separation distance, the percent of lost individuals found decreases by 0.5%. The units are percent per foot.
5Step 5: Find Intercepts
The vertical intercept \((d = 0)\) can be found by setting \(d = 0\) in the equation: \(P = -0.5(0) + 100 = 100\). This means that if searchers are together \((0\, \text{ft distance})\), 100% of lost individuals are found. The horizontal intercept \((P = 0)\) is found by setting \(P = 0\) and solving for \(d\): \(0 = -0.5d + 100\), resulting in \(d = 200\) feet. This means that at 200 ft separation, the effectiveness of the search drops to 0%.
Key Concepts
Slope InterpretationInterceptsRate of Change
Slope Interpretation
When we talk about the slope in a linear function, we're discussing how one variable changes in response to another. Imagine the slope as a number that tells you how steep a line is. In our exercise, the slope is -0.5. This is a crucial part of understanding linear functions as it describes the relationship between the percent of individuals found, \(P\), and the separation distance, \(d\), of the searchers.
The negative sign means there is a decrease: as the separation distance between searchers increases, the percent of people found decreases. Specifically, the slope of -0.5 suggests that for every 1 foot increase in distance, the percentage of individuals found decreases by 0.5%.
The negative sign means there is a decrease: as the separation distance between searchers increases, the percent of people found decreases. Specifically, the slope of -0.5 suggests that for every 1 foot increase in distance, the percentage of individuals found decreases by 0.5%.
- Units: The slope here is measured in percent per foot. This means for each additional foot searchers are separated, there is a 0.5% decrease in the percent found.
- Interpreting the Negative Slope: A negative slope indicates a losing rate in this context—fewer hikers are found when searchers are spread farther apart.
Intercepts
Intercepts are the points where the line crosses the axes on a graph. In our case, we have a vertical and a horizontal intercept, and they help us understand what happens in extreme cases.
First, let's discuss the vertical intercept, which occurs when \(d = 0\). Simply put, this means the searchers are standing right next to each other. When we plug \(d = 0\) into our linear equation \(P = -0.5d + 100\), we find \(P = 100\). This tells us that if there is no distance between searchers, they find 100% of the lost individuals. The vertical intercept here is 100% efficiency.
Next is the horizontal intercept, found where \(P = 0\). Solving the equation \(0 = -0.5d + 100\) gives us \(d = 200\). So, at a distance of 200 feet between them, the searchers find 0% of the lost individuals. This horizontal intercept illustrates that past a certain distance, the search team becomes ineffective.
First, let's discuss the vertical intercept, which occurs when \(d = 0\). Simply put, this means the searchers are standing right next to each other. When we plug \(d = 0\) into our linear equation \(P = -0.5d + 100\), we find \(P = 100\). This tells us that if there is no distance between searchers, they find 100% of the lost individuals. The vertical intercept here is 100% efficiency.
Next is the horizontal intercept, found where \(P = 0\). Solving the equation \(0 = -0.5d + 100\) gives us \(d = 200\). So, at a distance of 200 feet between them, the searchers find 0% of the lost individuals. This horizontal intercept illustrates that past a certain distance, the search team becomes ineffective.
- Vertical Intercept: \(d = 0\), \(P = 100\%\).
- Horizontal Intercept: \(P = 0\), \(d = 200\; \text{ft}\).
Rate of Change
The rate of change is the backbone of any linear function. It essentially tells us how one variable is affected when another variable changes. In the context of our linear relationship, it's the amount by which the percentage of lost individuals found, \(P\), changes as the separation distance between searchers, \(d\), is altered.
In simpler terms, it's the same as the slope, which we've calculated as \(-0.5\). This rate of change means every 1 foot increase in separation distance results in a \(-0.5\) reduction in the percentage of people found. It is a constant value, reflecting a consistent drop in efficiency over increasing distance.
In simpler terms, it's the same as the slope, which we've calculated as \(-0.5\). This rate of change means every 1 foot increase in separation distance results in a \(-0.5\) reduction in the percentage of people found. It is a constant value, reflecting a consistent drop in efficiency over increasing distance.
- Consistency of Rate of Change: This is consistent with the concept of linearity, as the rate remains the same regardless of where you are on the line.
- Real-world implication: This rate of change helps teams to strategize the most efficient distance between searchers to maximize the potential of finding lost individuals.
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