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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

Solve each problem. Maximum Number of Mosquitos The number of mosquitos, \(M(x),\) in millions, in a certain area of Florida depends on the June rainfall, \(x,\) in inches. The function $$ M(x)=10 x-x^{2} $$ models this phenomenon. Find the amount of rainfall that will maximize the number of mosquitos. What is the maximum number of mosquitos?

Solve each problem.Simple Interest Simple interest varies jointly as principal and time. If \(\$ 1000\) invested for 2 yr earned \(\$ 70\), find the amount of interest earned by \(\$ 5000\) for 5 yr.

If \(c\) and \(d\) are complex numbers, prove each statement. (Hint: Let \(c=a+b i\) and \(d=m+n i\) and form all the conjugates, the sums, and the products.) \(\bar{a}=a\) for any real number \(a\)

Find all complex zeros of each polynomial function. Give exact values. List multiple zeros as necessary. $$f(x)=x^{4}-8 x^{3}+29 x^{2}-66 x+72$$

Solve each problem. Automobile Stopping Distance Selected values of the stopping distance \(y,\) in feet, of a car traveling \(x\) miles per hour are given in the table. (a) Plot the data. (b) The quadratic function $$ f(x)=0.056057 x^{2}+1.06657 x $$ is one model that has been used to approximate stopping distances. Find and interpret \(f(45)\) (c) How well does \(f\) model the car's stopping distance?$$\begin{array}{c|c} \begin{array}{c} \text { Speed } \\ \text { (in mph) } \end{array} & \begin{array}{c} \text { Stopping Distance } \\ \text { (in feet) } \end{array} \\ \hline 20 & 46 \\ \hline 30 & 87 \\ \hline 40 & 140 \\ \hline 50 & 240 \\ \hline 60 & 282 \\ \hline 70 & 371 \\ \hline \end{array}$$
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