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Chapter 3: Problem 3

Write each formula as an English phrase using the word varies or proportional. \(r=\frac{d}{t},\) where \(r\) is the speed when traveling \(d\) miles in \(t\) hours?

Short Answer

Expert verified
Speed varies directly with distance and inversely with time.

Step by step solution

01

Identify the relationship

Examine the formula provided, which is given as \(r = \frac{d}{t}\), where \(r\) represents speed, \(d\) represents distance, and \(t\) represents time.
02

Interpret the formula

Understand that the formula describes how speed (\(r\)) is calculated by dividing distance (\(d\)) by time (\(t\)).
03

Express the Relationship in Words

The phrase 'r is the speed when traveling d miles in t hours' can be written using the word 'varies' or 'proportional' to describe the relationship. Here, speed (r) is directly proportional to distance (d) and inversely proportional to time (t).

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Key Concepts

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

speed calculation
Speed is a measure of how quickly something is moving. It's a fundamental concept in physics and everyday life. You can calculate speed using the formula: \( r = \frac{d}{t} \), where \( r \) is speed, \( d \) is distance, and \( t \) is time. This means that to find the speed, you divide the distance traveled by the time it took to travel that distance.
For example:
  • If you travel 100 miles in 2 hours, your speed is \( r = \frac{100 \text{ miles}}{2 \text{ hours}} = 50 \text{ miles per hour} \).

Speed tells us how fast we are going, and knowing how to calculate it can help us plan trips, estimate arrival times, and understand motion better.
direct proportionality
Direct proportionality means that as one quantity increases, the other also increases at a constant rate. In the context of speed, the formula \( r = \frac{d}{t} \) shows us that speed (\( r \)) is directly proportional to distance (\( d \)).
This implies:
  • If you increase the distance, keeping time constant, the speed increases.
  • For example, if you travel double the distance in the same amount of time, your speed also doubles.

This direct relationship is key to understanding many real-world scenarios such as driving, where driving a longer distance at a constant speed takes more time.
inverse proportionality
Inverse proportionality means that as one quantity increases, the other decreases at a constant rate. The formula \( r = \frac{d}{t} \) shows speed (\( r \)) as inversely proportional to time (\( t \)).
This implies:
  • If you increase the time, keeping distance constant, the speed decreases.
  • For example, if you take twice the time to travel the same distance, your speed is halved.

Understanding inverse proportionality helps in planning and optimizing travel, work schedules, and various other timed activities. It helps illustrate how increasing the allowed time for a task reduces the speed required to complete it.

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Most popular questions from this chapter

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Solve each problem. AIDS Cases in the United States The table* lists the total (cumulative) number of AIDS cases diagnosed in the United States through \(2007 .\) For example, a total of \(361,509\) AIDS cases were diagnosed through 1993 (a) Plot the data. Let \(x=0\) correspond to the year 1990 . (b) Would a linear or a quadratic function model the data better? Explain. (c) Find a quadratic function defined by \(f(x)=a x^{2}+b x+c\) that models the data. (d) Plot the data together with \(f\) on the same coordinate plane. How well does \(f\) model the number of AIDS cases? (e) Use \(f\) to estimate the total number of AIDS cases diagnosed in the years 2009 and 2010 (f) According to the model, how many new cases were diagnosed in the year \(2010 ?\) $$\begin{array}{c|c||c|c} \text { Year } & \text { AIDS Cases } & \text { Year } & \text { AIDS Cases } \\\ \hline 1990 & 193,245 & 1999 & 718,676 \\ \hline 1991 & 248,023 & 2000 & 759,434 \\ \hline 1992 & 315,329 & 2001 & 801,302 \\ \hline 1993 & 361,509 & 2002 & 844,047 \\ \hline 1994 & 441,406 & 2003 & 888,279 \\ \hline 1995 & 515,586 & 2004 & 932,387 \\ \hline 1996 & 584,394 & 2005 & 978,056 \\ \hline 1997 & 632,249 & 2006 & 982,498 \\ \hline 1998 & 673,572 & 2007 & 1,018,428 \\ \hline \end{array}$$
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