A GPS device tracks the elevation

Question

A GPS device tracks the elevation $E$ (in feet) of a hiker walking in the mountains. The elevation $t$ hours after beginning the hike is given in the figure.

Find the slope of the secant line that passes through points $A$ and $B$ . Interpret your answer as an average rate of change over the interval $1\leq{t}\leq{3}$ .

Accepted Answer

Hi, everyone. Let's take a look at this practice problem dealing with finding the slope of a line. This problem says a car travels along a straight road and its distance D in miles from the starting point is recorded every hour. The distance traveled two hours after the journey began is recorded. A second line that passes through the points L and M is drawn, determine the slope of this line and interpret it as an average rate of change over the interval of two is less than or equal to T is less than or equal to four. Below the question. We're given a graph where we have our distance D on the Y axis and our time T along the X axis. In the graph, there is a curve and there are points labeled on the curve at every one hour intervals. There are also two points labeled L and M on the graph. The point L has a value of two comma 567 and the point M has a value of four comma 643. We're also given four possible choices as our answers. For choice. A we have 38 MPH. The car moves away from the starting point at an average rate of 38 MPH. For choice. B we have minus 38 MPH. The car moves towards the starting point at an average rate of 38 MPH. For choice. C we have 36 MPH. The car moves away from the starting point at an average rate of 36 MPH. And for choice D, we have minus 36 MPH. The car moves towards the starting point at an average rate of 36 MPH. Now, we're asked to determine the slope of the secret line that is connecting points L and M. So in order to find the slope of this line, we're actually going to use our coordinates for points L and M. So we need to actually determine what those are. Now for point L, this occurs when our time T is equal to two hours and the value of our distance in that case, which will write as DF two that is gonna be equal to 567. And I see that just by looking at the value of point L on the graph, we do the same thing for point M. It occurs at T is equal to four hours and it's Y value which will label as D of four, it's gonna be equal to, if I look at the value that's given on the graph that is 643 miles. So I need to use these two points to determine the slope. As you recall your equation for the slope. That is that your slope is equal to your change in Y divided by your change in X. So our slope M it's gonna be equal to our change in Y which is gonna be the 643 minus the 567. And that's gonna be divided by our change in X which is four minus two. We plug all those values into our calculator. We'll get M is equal to 38. And this has units of MPH since the units for our numerator were in miles and the units for our denominator were in hours. So this is gonna be the first half of our answer to this question. For the second half, we need to interpret this as an average rate over the interval going from 2 to 4. And in this case, that's just going from the point L to point M. So our average rate is going to be the 38 MPH and this rate is actually positive. So that means that our distance D is actually increasing with time. So this means that we're actually moving away from the starting point at an average rate of 38 MPH. So with that interpretation, we can look at our answer choices and the correct answer choice actually corresponds to answer choice. A so I hope that this explanation has been useful and I'll see you in the next video.

Key Concepts

Slope of a Secant Line

Average Rate of Change

Function Representation

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