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Slope in Context

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Today we're going to talk about slope in the context of real world situations. Remember, slope is just the vertical distance divided by the horizontal distance between two points. So we're going to do some examples to show you how the slope can be thought of, in terms of the variables in real world situations.

So for my first example, suppose Anne can run 10 miles in 1 and 1/2 hours, or 1.5 hours. We want to know what her average speed was during that run. So we're going to use the relationship between distance, rate, and time. And we're going to write an equation in terms of the rate, which would be equal to your distance traveled, over the time it took you to travel that distance.

So for our problem, that means our rate is going to be equal to 10 miles over 1.5 hours. Dividing 10 by 1.5, I can see that my rate is going to be equal to approximately 6.7 miles per hour. So even though Anne may have ran at varying speeds during her run, the rate of 6.7 miles represents her average speed, which would give you the constant race that she would need to run at to cover the 10 miles in 1.5 hours.

Let's do another example. So let's look at a graph that could represent data of Anne's journey on her run. On this graph, I've got time plotted on my horizontal, or x-axis-- time in hours. And on my vertical axis, or my y-axis, I've got distance in miles. So I plotted points. For example, at time 0, she's gone a distance of 0. After a quarter of an hour, or 15 minutes, she had gone let's say a little bit more than two miles. And we continued on like this until we got to the end of her run, 1.5 hours, she had traveled 10 miles.

So this represents the fact that Anne did not run at a constant speed throughout her journey. In fact, she ran a lot faster in the beginning than she did at the end. But we determined that her average speed was about 6.7 miles per hour. So we could plot that on our same graph. So we know again that she started at 0. At time 0, her distance was 0. And by the end of her run, 1.5 hours, she had gone 10 miles. And since we are finding the average, we'll represent that with a straight line. And so that we can see that the intersection point for her actual run, at the beginning of her run, and when we're plotting the points for her average distance, is the same.

We also see that the intersection point at the end of her run, both for when we're looking at her average speed and when we're looking at her actual time and distance data, the intersection point for the end of her run is also the same.

Another thing to note is that the slope of this line, that represents her average speed, is going to be her rate, or her average speed. So if we were to calculate the slope between these two points, we would see that the slope would be equal to 10 over 1.5, which again would just be equal to approximately 6.7, which is her average speed for the run.

All right. So now let's suppose we have Kristen, who's on a hike. And while hiking, she traveled a total horizontal distance of 5 miles, and she traveled a total vertical distance up the mountain of 2,500 feet. So let's figure out what her average rate of change, of her vertical distance over her horizontal distance, would be.

So we know that she traveled a total of 2,500 vertical feet. And that the same amount of time, on her hike, she traveled a total of 5 horizontal miles. So simplifying this, we know that 2,500 divided by 5 is going to give us 500. And the units will be 500 vertical feet for every 1 horizontal mile. So this average rate of change can also be represented as the slope between the two points, if we were to graph where she started at the beginning of her height and where she ended. So let's see what that would look like.

So I know by looking at the graph, with the horizontal distance in miles on my x-axis and the vertical distance in feet on my y-axis, that at the beginning of her hike, her horizontal and vertical distance are both 0. And at the end of her hike, her horizontal distance was 5 miles and her vertical distance was 2,500 feet. So I can represent these two points as a coordinate pair of points. This first one would be 0,0, and my second one would be 5, 2500.

So substituting those values into a formula that we know for slope, I know that I want to find the change in the vertical distance, so that's going to be 2,500 minus 0, over the change in my horizontal distance. So that's going to be 5 minus 0. Simplifying on the top and on the bottom, this becomes 2,500 over 5, which gives me the same value of 500 for the average rate of change between the vertical distance and the horizontal distance traveled.

All right. So now let's suppose that we have three friends-- Brandon, D'Andre, and Mason. And they are monitoring their weight over the course of the year. They weigh themselves at the beginning of the year, and at the end of a year, or 12 months, and created a graph to show the data points at the beginning of the year and at the end of the year.

So Brandon started at about 125 pounds at the beginning of the year and ended at the end of the year at the same weight, 125 pounds. So his change in his average weight was 0. He started and ended in the same place, and we can see from the graph that that is a horizontal line, which we know horizontal lines have slopes of 0. D'Andre started at the beginning of the year at 150 pounds, and ended at the end of the year at 200 pounds. So he gained 50 pounds over the course of the year. So we can see that an increase in weight corresponds to a slope that is positive, it is going upwards, or towards positive infinity on our y-axis. And our y-axes.

And then finally, we have Mason, who started at 350 pounds at the beginning of the year and ended at about 275 pounds at the end of the year. So because he lost weight over the year, we know that his average change in weight is going to be negative. And we can see that, from our graph, that it corresponds to the slope of the line that is being negative, because it is heading towards negative infinity on our y-axis.

So let's go over our key points from today. You can use the total time and total distance traveled to calculate an object's average speed. The average speed describes a constant rate the object traveled to cover a distance over a certain period of time. And the slope of a line that represents an object's type and distance will equals the average rate of change or speed of the object.

So I hope that these key points and examples helped you understand a little bit more about using the slope in the context of real world problems. Keep using your notes and keep on practicing, and soon you'll be a pro. Thanks for watching.