Question

making sense of slope - item 32532
the graph shows the motion of two cars starting at different places on a highway. their speeds can be compared by comparing the steepness of the graphed lines.
use the drop - down menus to complete the statements.
car a travels miles in 3 hours. car b travels miles in 2 hours.
car as speed is miles per hour. car bs speed is miles per hour.
car b traveled car a. the graph for car b is the graph for car a.

Explanation:

Step1: Find distance for Car A

From the graph, when \(x = 3\) (time in hours), for Car A, \(y=100\) (distance in miles).

Step2: Find distance for Car B

From the graph, when \(x = 2\) (time in hours), for Car B, \(y = 150\) (distance in miles).

Step3: Calculate speed of Car A

Speed \(v=\frac{\text{distance}}{\text{time}}\). For Car A, \(v_A=\frac{100 - 25}{3}=\frac{75}{3}=25\) miles per hour (using two - point \((0,25)\) and \((3,100)\) on Car A's line. The formula for slope \(m=\frac{y_2 - y_1}{x_2 - x_1}\), and speed is the slope of the distance - time graph).

Step4: Calculate speed of Car B

For Car B, using the point \((0,0)\) and \((2,150)\), speed \(v_B=\frac{150-0}{2}=75\) miles per hour (since slope \(m = \frac{y_2 - y_1}{x_2 - x_1}\) and for distance - time graph, slope is speed).

Step5: Compare speeds and steepness

Since \(v_B=75>v_A = 25\), Car B is faster. In a distance - time graph, a steeper line (greater slope) represents a higher speed.

Answer:

Car A travels \(100\) miles in 3 hours. Car B travels \(150\) miles in 2 hours.
Car A’s speed is \(25\) miles per hour. Car B’s speed is \(75\) miles per hour.
Car B traveled faster than Car A. The graph for Car B is steeper than the graph for Car A.