Question

car x travels 186 miles in 3 hours.
a. write the equation of the line that describes the relationship between distance and time. use x for the time in hours and y for the distance in miles.
b. what is the graph that represents the relationship between distance and time for car x? explain.
if a car travels for 0 hours, it will travel \\(\square\\) mile(s). this means it will pass through the point \\(\square\\). use the slope to move 3 units to the right of the origin and \\(\square\\) unit(s) up to find the point \\(\square\\) that is used to graph the relationship.

Explanation:

Part a: Equation of the Line

Step1: Find the slope (rate of speed)

The speed (slope \( m \)) is distance divided by time. So \( m = \frac{186}{3} = 62 \).

Step2: Determine the y-intercept

When time \( x = 0 \) (0 hours), distance \( y = 0 \) (since a car moving for 0 hours travels 0 miles). So the y-intercept \( b = 0 \).

Step3: Write the linear equation

The slope - intercept form of a line is \( y = mx + b \). Substituting \( m = 62 \) and \( b = 0 \), we get \( y = 62x \).

The relationship between distance and time for a car moving at a constant speed is a linear relationship. The equation \( y = 62x \) is a proportional relationship (since \( b = 0 \)), so the graph should be a straight line passing through the origin \((0,0)\). When \( x = 3\) (3 hours), \( y=62\times3 = 186\) miles. So the graph should pass through \((0,0)\) and \((3,186)\). Looking at the given graphs, the second graph (with x - axis as time in hours and y - axis as distance in miles) has the x - axis as time and y - axis as distance, and it passes through the origin and has a positive slope, which matches the relationship \( y = 62x \). The first graph has x - axis as distance and y - axis as time, which is an inverse relationship (not what we want here since distance should increase with time). So the second graph (with Time (hr) on x - axis and Distance (mi) on y - axis, passing through \((0,0)\) and \((3,186)\) etc.) represents the relationship.

• If a car travels for 0 hours, it will travel 0 mile(s). This is because distance \( y=62\times0 = 0\).
• This means it will pass through the point \((0,0)\).
• The slope \( m = 62\), which means for every 1 unit we move to the right (in x - direction, time), we move 62 units up (in y - direction, distance). If we move 3 units to the right of the origin (x = 3), we move \( 62\times3=186\) units up.
• The point we find is \((3,186)\).

Answer:

\( y = 62x \)

Part b: Graph Explanation