ERAST Pathfinder:

Battery Depletion and Piecewise Linear Graphing


Author: Robin A. Ward, California Polytechnic State University-San Luis Obispo


Audience: algebra 2 students


Mathematical Topics: linear graphing, computing slope and y-intercepts


Rationale: The NCTM Standards recommend that students be able to model real-world phenomena with a variety of functions as well as be able to translate among tabular, symbolic, and graphical representations of functions. The Standards also emphasize making links across the curriculum. This activity allows students to see a real-life application of linear graphing while gaining experience with the various representations of functions.


Background: Unlike most conventional aircraft, Pathfinder flies without an onboard human pilot. Instead, it is controlled remotely from a ground station. The upper surface of the aircraft's 100- foot wing is covered almost completely by thin solar powered panels, which collect sunlight. These solar arrays can provide as much as 7,200 watts of power. Pathfinder converts energy from the sun into electricity, which turns six small motors with propellers. Slowing down or speeding up these individual propellers allows Pathfinder to make turns, since it does not have the control surfaces of typical aircraft. Because Pathfinder is solar-powered, it can stay aloft for a week or more.

Despite being solar powered, Pathfinder is also battery powered. Pathfinder needs to have a back-up battery for those incidences where the aircraft does not receive any solar-powering. Recall that because Pathfinder can fly continually for a week or more that the aircraft will encounter periods of no sunlight.


The Activity:

  • On a recent mission in August of 1997, Pathfinder collected the following data concerning its battery state of charge. During the first ten hours of flight, the battery state of charge was recorded to be 100%. Over the course of the next three hours, the charge decreased at a steady rate of 7.5% per hour.

  • Using the information provided above, the students will first construct a table of values showing Pathfinder's battery state of charge as time passed. Based on the information provided, the following table of (x, y) coordinate pairs can be generated (where the x-coordinate is time and the y-coordinate is the battery state of charge).



  • Once the table is constructed, ask students to predict what the data might look like if they were to plot the battery state of charge (y-axis) versus time (x-axis). Then, using their table of values, students will construct a graph representing the battery state of charge (y-axis) versus time (x-axis).



  • Next, students will algebraically express the two linear equations represented in this piecewise (two linear pieces) graph.

  • The equation of the horizontal piece is y = 100, since every y-value (for the first ten data points) is 100, regardless of the x-coordinate.

  • The equation of the "slanted" piece is y = -7.5x + 175. This equation was obtained as follows. The battery is losing its charge at a steady rate of 7.5% each hour. Thus, the slope, or rate of change, is -7.5. Next, we plug the value of slope into the slope-intercept form of the line, y = mx + b, as shown below. Recall that the m in the equation represents the slope.

    • y = mx + b

    y = -7.5x + b

  • To solve for b, the y-intercept, choose any of the (x, y) ordered pairs that are points for this portion of the graph. Let's choose (12, 85). Plugging 12 in for x and 85 in for y, we get:

    • 85 = -7.5 (12) + b

    or

    85 = -90 + b

    or

    175 = b.

  • Now that we have a value for b, the y-intercept, we can plug this value, along with the value for slope, into the slope-intercept form of a line and get:

    • y = -7.5x + 175.

  • Ask students if the value of 175 for the y-intercept makes sense. Point out to students that if we were to extend that slanted line upwards, it would indeed intersect at 175 on the y-axis; hence the name, "y-intercept."


  • Now that students have found the linear equations for both pieces of the graph, challenge students to devise a way of representing both equations succinctly. One possibility is shown below:


    Enrichment Activities: Present the following scenario to students: Suppose you are a NASA Dryden engineer remotely piloting Pathfinder from the ground. Knowing that Pathfinder's battery is losing 7.5% of its charge with every passing hour, can you predict the battery state of charge after 20 hours of flight?

    When will the battery have no power at all left?

    If you were a NASA Dryden engineer and wanted to insure that Pathfinder would make a safe landing back on Earth, when would you land the aircraft? How low would you let the battery run down? Promote a classroom discussion and encourage students to justify their reasoning. Or, ask students to write about this in their journals and then share their thoughts with the class.

    Students may observe and plot other Pathfinder data that also shows a linear relationship. Allow students to discover the linear relationship between altitude and temperature.

    Students may also observe and plot other Pathfinder data that does not show a relationship. Come analyze altitude and wind speed data collected by Pathfinder during one of its flights.


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    • Funded by the NASA Dryden Flight Research Center


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