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Chapter 2: Problem 11

A bug is crawling horizontally along the wall at a constant rate of 5 inches per minute. You first notice the bug when it is in the corner of the room, behind your music stand. a. Define variables and write an equation that relates time (in minutes) to distance traveled (in inches). (a) b. What is the constant of variation of this direct variation relationship, and what does it represent? c. How far will the bug crawl in \(1 \mathrm{~h}\) ? d. How long would you have to practice playing your instrument before the bug completely "circled" the \(14-\mathrm{ft}-\mathrm{by}-20\)-ft room? (a) e. Draw a graph that represents this situation.

Short Answer

Expert verified
a) Equation: \( d = 5t \); b) Constant: 5 in/min; c) 300 inches; d) 163.2 minutes; e) Graph: line through origin with slope 5.

Step by step solution

01

Define Variables

Let's define the variables: Let \( t \) represent time in minutes and \( d \) represent the distance traveled in inches. The bug crawls at a constant rate of 5 inches per minute.
02

Write the Equation

Since the distance traveled \( d \) is directly proportional to time \( t \), we can write the equation as \( d = 5t \). Here, 5 is the rate (or constant of variation) at which the bug moves.
03

Identify the Constant of Variation

The constant of variation in this direct variation relationship is 5 inches per minute. This represents the speed of the bug, or how far it travels in one minute.
04

Calculate Distance in One Hour

To find out how far the bug will crawl in an hour, substitute \( t = 60 \) minutes into the equation \( d = 5t \):\[d = 5 \times 60 = 300 \text{ inches}\]Thus, the bug will crawl 300 inches in one hour.
05

Calculate Time to Circle the Room

First, convert the room's dimensions to inches: 14 ft = 168 inches and 20 ft = 240 inches. The perimeter of the room is:\[2(168) + 2(240) = 816 \text{ inches}\]Now, use the equation \( d = 5t \) to find the time when \( d = 816 \):\[816 = 5t \implies t = \frac{816}{5} = 163.2 \text{ minutes}\]Thus, you would have to practice for 163.2 minutes.
06

Graph the Relationship

Plot the graph with time \( t \) on the x-axis and distance \( d \) on the y-axis. The graph will be a straight line passing through the origin (0,0) with a slope of 5, showing that distance increases by 5 inches for each additional minute. It should extend beyond 60 minutes to show how the bug continues to crawl over time.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Constant of Variation
When discussing direct variation, the constant of variation is a fundamental concept. In the context of our exercise, the bug crawls at a rate of 5 inches per minute. This rate is what we call the 'constant of variation'. It shows how the distance traveled by the bug changes in relation to time. In technical terms, the distance is directly proportional to time. The equation representing this relationship is given by \( d = 5t \), where \( d \) is the distance in inches and \( t \) is the time in minutes. The constant of variation, 5, tells us that for every minute, the bug moves 5 inches. This steady and predictable rate is what defines direct variation, making it a straightforward concept in mathematics.
Distance-Time Equation
The distance-time equation is a powerful tool. It shows how much distance is covered in a given time and is particularly useful in solving real-world problems such as our bug crawling scenario. Let's break down the equation \( d = 5t \). Here, \( d \) refers to the distance the bug has traveled in inches, and \( t \) represents the time in minutes.
The equation implies that the bug's movement is consistent and predictable. With the distance-time equation, you can calculate how far the bug travels over any specific duration. For instance, substituting \( t = 60 \) (for an hour) into the equation gives \( d = 5 \times 60 = 300 \) inches. Thus, you can predict distances for varying times, making the equation a crucial part of solving distance-related problems.
Graphing Linear Equations
Graphing linear equations helps visualize relationships like the one between distance and time for our bug. The graph of the equation \( d = 5t \) is a straight line. This line passes through the origin (0,0) since, initially, at zero minutes, the bug has traveled zero inches.
On the graph, the x-axis will represent time \( t \), while the y-axis will reflect distance \( d \). With a constant slope of 5, this indicates that each minute corresponds to a travel distance of 5 inches. By plotting various points, like \((1, 5), (2, 10)\), and so on, you can see how linearly the distance increases with time. Such graphs offer a visual interpretation of equations, making complex relationships easier to understand and analyze.

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