Question Video: Finding the Slopes of Straight Lines from Graphs in a Real-World Context | Nagwa Question Video: Finding the Slopes of Straight Lines from Graphs in a Real-World Context | Nagwa

Question Video: Finding the Slopes of Straight Lines from Graphs in a Real-World Context Mathematics

The graph shows the distance Amelia traveled over her 2 hour bike ride. Which of the following is true? [A] She traveled at a constant speed of 4 miles per hour for the last hour. [B] She traveled at a constant speed of 10 miles per hour for the entire ride. [C] She traveled at a constant speed of 8 miles per hour for the last hour. [D] She traveled at a constant speed of 7 miles per hour for the entire ride.

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Video Transcript

The graph shows the distance Amelia traveled over her two-hour bike ride. Which of the following is true? A) She traveled at a constant speed of four miles per hour for the last hour. B) She traveled at a constant speed of 10 miles per hour for the entire ride. C) She traveled at a constant speed of eight miles per hour for the last hour. Or D) she traveled at a constant speed of seven miles per hour for the entire ride.

We can see from the graph that the 𝑥-axis represents the time in hours and the 𝑦-axis represents the distance in miles. The speed or velocity in any distance-time graph can be calculated by dividing the change in distance between any two points by the change in time. If the graph is a straight line for the entire journey, then they will be traveling at a constant speed. We can see from the graph that three parts of the journey have different slopes or gradients. This means that, during these three parts, Amelia will be traveling at different speeds.

We can therefore rule out options B and D, as these stated that she traveled at a constant speed for the entire ride. This is not the case as she will have traveled at three different speeds. Both of the other statements relate to the last hour of Amelia’s journey. This occurs between the two points 𝐴 and 𝐵 on the graph. We can calculate the slope between any two points on a graph by using the following formula, 𝑦 two minus 𝑦 one over 𝑥 two minus 𝑥 one. This is the change in 𝑦-coordinates over the change in 𝑥-coordinates, in this case the change in the distance over the change in the time.

Point 𝐴 has coordinates one, 10 and point 𝐵 has coordinates two, 14. The 𝑦-coordinates or distances here are 14 and 10. The corresponding 𝑥-coordinates are two and one. 14 minus 10 is equal to four and two minus one is equal to one. This means that the slope of the line between points 𝐴 and 𝐵 is four. We could also have worked this out by drawing a right-angled triangle on the graph. We can see here that the distance has risen from 10 to 14. And the time has gone from one hour to two hours. Four divided by one is equal to four. So once again, the slope equals four.

As the slope in a distance-time graph is equal to the speed, we can conclude that the speed in the last hour was four miles per hour. This rules out option C and therefore option A is correct. Amelia traveled at a constant speed of four miles per hour for the last hour.

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