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Chapter 1: Problem 9

A blue plane flies across the country at a constant rate of 400 miles per hour. a. Is the relationship between hours of flight and distance traveled linear? b. Write an equation and sketch a graph to show the relationship between distance and hours traveled for the blue plane. c. A smaller red plane starts off flying as fast as it can, at 400 miles per hour. As it travels it burns fuel and gets lighter. The more fuel it burns, the faster it flies. Will the relationship between hours of flight and distance traveled for the red plane be linear? Why or why not? d. On the axes from Part b, sketch a graph of what you think the relationship between distance and hours traveled for the red plane might look like.

Short Answer

Expert verified
a. Yes. b. The equation is \(d = 400h\). c. No, because the speed increases over time. d. A curve that gets steeper over time.

Step by step solution

01

Determine if the Relationship is Linear (part a)

To determine if the relationship between hours of flight and distance traveled for the blue plane is linear, observe the constants in the problem. The plane flies at a constant rate of 400 miles per hour which means there is a direct proportionality between time and distance. Therefore, the relationship is linear.
02

Write the Equation (part b)

Write the linear equation for the blue plane using the formula for distance, which is distance = rate 脳 time. Here, the rate is 400 miles per hour. Thus, the equation becomes:\[ d = 400h \]where \(d\) is the distance traveled in miles, and \(h\) is the number of hours traveled.
03

Sketch the Graph (part b)

To sketch the graph, plot a line where the independent variable (hours traveled, \(h\)), is on the x-axis, and the dependent variable (distance traveled, \(d\)) is on the y-axis. Starting from the origin (0,0), draw a line with a slope of 400. This line represents the linear relationship given by the equation \(d = 400h\).
04

Determine if the Relationship is Linear for the Red Plane (part c)

For the smaller red plane, the speed is not constant; it increases over time as it burns fuel. This means the distance traveled per unit time is not constant, and hence the relationship between hours of flight and distance traveled is not linear.
05

Sketch the Graph for the Red Plane (part d)

To sketch the graph, start at the origin (0,0). Since the red plane starts at 400 miles per hour and increases speed over time, the graph will be a curve that starts with a slope of 400 and gets steeper as time increases. This might appear as an upward-sloping curve that gets progressively steeper.

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Key Concepts

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

constant rate
A 'constant rate' means something happens at the same speed all the time. For example, the blue plane in the exercise flies at a constant rate of 400 miles per hour.

When you hear 'constant rate', think of steady and unchanging speed. This is important in algebra because it helps us form linear equations. If a car travels at a constant speed of 60 miles per hour, in 2 hours it will travel 120 miles. The rate (60 miles per hour) stays the same.

In the exercise, the blue plane flies at a constant rate of 400 miles per hour. This means for each hour it flies, it always covers 400 miles, without changing.
linear equation
A 'linear equation' is an equation that makes a straight line when you graph it. It's called 'linear' because it forms a line. The general form of a linear equation is: y = mx + b.

In this formula, 'y' is the dependent variable, 'x' is the independent variable, 'm' is the slope, and 'b' is the y-intercept (where the line crosses the y-axis).
In the exercise, the linear equation for the blue plane is:
d = 400h. Here, 'd' is the distance traveled and 'h' is the hours flown. The slope 'm' is 400 because the plane travels 400 miles each hour. There's no y-intercept (b=0) because the line starts from the origin (0,0), when the plane hasn鈥檛 flown any hours yet.

Linear equations help you predict how one variable will change when another variable changes. In this case, if you know how many hours the plane flies, you can use the linear equation to find out how far it goes.
distance-time graph
A 'distance-time graph' shows how distance changes over time. The horizontal axis (x-axis) usually represents time, while the vertical axis (y-axis) represents the distance.

For the blue plane, the graph starts at the origin (0,0) and climbs upwards in a straight line. This indicates a constant speed. If you plot points, like at 1 hour (400 miles), 2 hours (800 miles), and 3 hours (1200 miles), you'll see they fall on a straight line.

The slope of this line is the speed of the plane 鈥� in this case, 400 miles per hour. As time increases by 1 hour, distance increases by 400 miles.

In the case of the red plane with variable rate, the distance-time graph forms a curve that gets steeper over time, showing increasing speed.
variable rate
A 'variable rate' means the speed or rate changes over time. Unlike 'constant rate', it is not steady.

In the exercise, the red plane's speed increases as it burns fuel. This creates a variable rate of speed because the plane goes faster the longer it flies.

When you have a variable rate, the relationship is not linear. The graph of such a situation is not a straight line. For the red plane, the graph starts like the blue plane's but quickly becomes steeper as time goes on.

Understanding the difference between constant and variable rates helps in predicting outcomes and forming accurate graphs in algebra.

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Most popular questions from this chapter

For Exercises 20鈥�28, answer Parts a and b. a. What is the constant difference between the \(y\) values as the \(x\) values increase by 1\(?\) b. What is the constant difference between the \(y\) values as the \(x\) values decrease by 2\(?\) $$ y=-2 x+12 $$

Three navy divers are trapped in an experimental submarine at a remote location. They radio their position to their base commander, calling for assistance and more oxygen. They can use the radio to broadcast a signal to help others find them, but their battery is running low. The base commander dispatches these three vehicles to help: 鈥� a helicopter that can travel 45 miles per hour and is 300 miles from the sub 鈥� an all-terrain vehicle that can travel 15 miles per hour and is 130 miles from the sub 鈥� a boat that can travel 8 miles per hour and is 100 miles from the sub Each vehicle is approaching from a different direction. The com- mander needs to keep track of which vehicle will reach the submarine next, so he can tell the sub to turn its radio antenna toward that vehicle. a. To assist the base commander, create three graphs on one set of axes that show the distance each vehicle is from the sub over time. Put time on the horizontal axis, and label each graph with the vehicle鈥檚 name. b. Use your graphs to determine when the commander should direct the submarine to turn its antenna towards each of the following: the helicopter, the all- terrain vehicle, and the boat. c. Write an equation for each graph that the commander could use to determine the exact distance \(d\) each vehicle is from the submarine at time \(h\).

For each equation, identify the slope and the y-intercept. Graph the line to check your answer. $$ y=x-3 $$

The table shows x and y values for a particular relationship. $$\begin{array}{|c|c|c|c|c|}\hline x & {6} & {3} & {1} & {2.5} \\ \hline y & {7} & {1} & {-3} & {0} \\ \hline\end{array}$$ $$\begin{array}{l}{\text { a. Graph the ordered pairs }(x, y) . \text { Make each axis scale from }-10} \\ {\text { to } 10 .} \\ {\text { b. Could the points represent a linear relationship? If so, write an }} \\ {\text { equation for the line. }} \\ {\text { c. From your graph, predict the } y \text { value for an } x \text { value of }-2 . \text { Check }} \\ {\text { your answer by substituting it into the equation. }}\end{array}$$ $$ \begin{array}{l}{\text { d. From your graph, find the } x \text { value for a } y \text { value of }-2 . \text { Check your }} \\ {\text { answer by substituting it into the equation. }} \\ {\text { e. Use your equation to find the } y \text { value for each of these } x \text { values: }} \\ {0,-- 1,-1.5,-2.5 . \text { Check that the corresponding points all lie on }} \\\ {\text { the line. }}\end{array} $$

Consider these tables of data for two linear relationships. $$\begin{array}{|c|c|}\hline & {\text { Rationship } 1} \\ \hline x & {y} \\\ \hline 1 & {4.5} \\ {2} & {6} \\ {3} & {7.5} \\ {4} & {9} \\\ \hline\end{array}$$ $$\begin{array}{|c|c|}\hline & {\text { Rlationship } 2} \\ \hline x & {y} \\\ \hline-3 & {1} \\ {-1} & {3} \\ {1} & {5} \\ {3} & {7} \\\ \hline\end{array}$$ a. Use the \((x, y)\) pairs in the tables to draw each line on graph paper. b. Find the slope of each line by finding the ratio \(\frac{\text { rise }}{\text { ran }}\) between two points. c. Did you use the tables or the graphs in Part b? Does it matter which you use? Explain. d. Check your results by finding the slope of each line again, using points different from those you used before.
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