Area = Distance?

Mitch Ehrman

7/4/09

Students are frequently reminded that the units for area and distance are very different. For example a square foot is not the same as linear foot. Today we will look at an example of how area can represent distance through the use of integrals. Though the definition of integrals will not be given in this lesson, but the general concept will be known.

Problems 1 and 2 are pretty simple, and I hope students will recognize that the shape of the first graph is a rectangle whose area is 1 square unit. Clearly, a car will travel 60 miles in one hour, but a short discussion will help them realize that a car travels one mile each minute when driving 60 miles per hour.

The second graph will be a triangle whose area is 0.5. They should recognize that they would travel half the distancein this case.

Problem 3 will be difficult for some students to follow. Where the equation is irrelevant (but good exploration for some of the more advanced students), but students need to realize that the geogebra graph has mixed units: on the x-axis, time is measured in MINUTES, while the y-axis shows the rate in miles per HOUR. The area of the trapezoid formed is 40 square units. To accommodate for the mixed units, students should divide their answer by 60 to see that the car travels 2/3 mile in one minute, following this acceleration/deceleration schedule.

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1) If you drive 60 miles per hour for one hour, how far have you traveled?

Graph y=60, on the range [0,1]. What shape do you have? What is its area?

2) Suppose you steadily increase speed from 0 to 60 mph for one hour. What shape would that graph look like? What is its area?

3) Now suppose you are driving the car so that you constantly accelerate to 60 mph for twenty SECONDS, you cruise at 60 mph for twenty SECONDS, and you constantly decelerate for twenty SECONDS. The motion of your car can be modeled by this piecewise function:

a)Take a look at the Geogebra worksheet.

b)What is its area?

c)Why does this number misrepresent the distance traveled?

d)There are a few ways to correct this problem. Determine one way that works, and explain your solution in a short (four or five sentence) paragraph.