Illuminations: On Top of the World

On Top of the World
If you were standing on the top of Mount Everest, how far would you be able to see to the horizon? In this lesson, students will consider two different strategies for finding an answer to this question. The first strategy is algebraic—students use data about the distance to the horizon from various heights to generate a rule. The second strategy is geometric—students use the radius of the Earth and right triangle relationships to construct a formula. Then, students compare the two different rules based on ease of use as well as accuracy.

Learning Objectives
 
By the end of this lesson, students will:
  • Generate a formula that can be used to predict the distance to the horizon from a given height above sea level.
  • Determine the distance to the horizon from the summit of Mount Everest (29,035 feet).

Materials
 
Top of the World Activity Sheet
Graphing calculators
Computers with Internet access

Instructional Plan
Ask students to tell their classmates about the tallest place they've ever been. If students remember, have them list the height above sea level of those locations, and ask them to describe what they saw from that height. In most cases, students will be able to describe how far they were able to see to the horizon. For reference, research the height above sea level of your current location, as well as the height above sea level of a few nearby attractions. If there is a well-known mountain nearby, find out the height of its summit. Should students have difficulty remembering details about places they've been, use the data about nearby locations to give students a sense of height above sea level.

Now ask students to imagine what it would be like to stand on the top of Mount Everest, which is 29,035 feet above sea level. Ask them, "How far do you think you'd be able to see to the horizon if you were standing at the summit of Mount Everest?" Allow students to speculate, but do not offer opinions about their estimates. Explain that the focus of this lesson is for them to attempt to answer that question.

Distribute the first page of the Top of the World activity sheet. Inform students that they should answer questions 1 and 2 fairly quickly; at this point, their guess should be based solely on a quick estimate, not on extensive research or calculations. Later in the lesson, they will have an opportunity to develop a general rule based on data provided.

On Top of the World Activity Sheet Top of the World Activity Sheet

After all students have completed the first page, conduct a brief discussion about their answers. However, do not judge or comment on their estimates. You may allow other students to offer comments, but you should only say that thinking about this estimate is a precursor to the activity that they will complete today. Then, divide the class into pairs or small groups; to half of these groups, distribute the sheet titled "Algebraic Approach," and to the other half of the groups, distribute the sheet titled "Geometric Approach." Allow the groups to work on their respective sheets, and then have groups with different sheets compare their results. They may be surprised by the results.

After students have completed the activity sheet, conduct a brief class discussion. Focus this discussion on the following questions:

  1. Using your formulas, determine how far a person standing at the top of Mount Everest would be able to see to the horizon. Compare the results generated by each formula.
  2. Consider a rocket 1000 miles (or 5,280,000 feet) above the Earth. Use both of your formulas to determine how far a person in this rocket would be able to see to the horizon. Do both formulas give the same result?

To conclude the lesson, allow students to explore the Distance to Horizon applet. This tool allows the user to adjust the height above sea level, and the distance to the horizon is calculated. The height can be adjusted to exactly 29,035 feet, the height of Mount Everest. At this height, the applet shows that the distance to the horizon is 208.8 miles. Students can use this value as a check of their results.

Distance to Horizon Applet Distance to Horizon Applet

Selected Solutions to the Answer Sheet

Question 4. A power regression (accessible through the PwrReg feature on TI calculators) gives the function of best fit. The regression corresponds to the function d = 1.218 · h0.501. (This result is very similar to a formula used by ship captains who wished to estimate the distance to a shore. To calculate the distance to the horizon, they estimated the value of d = (1.5h)0.5.)

Questions 5 and 9. The power function above yields a result of 208.9 miles, whereas the formula that results from the geometric approach yields a result of 208.8 miles.

Question 7. The Pythagorean theorem can be used to generate an equation, but it must be remembered that the distance to the horizon is measured in miles, while the height above sea level is measured in feet. Consequently, a conversion needs to be done. Let n = h/5280, which is the height above sea level in miles. Then, the resulting equation is (3963 + n)2 = 39632 + d2.

Question 8. Solving for d then gives d = (7926n + n2)0.5.

Questions for Students
 
How can the distance to the horizon be determined based on the height above sea level?
[The sight line from a certain height to the horizon is tangent to the Earth and therefore forms a right angle with the radius to the point of tangency, as shown below.

Consequently, a right triangle is formed, so the Pythagorean theorem can be applied. It must be known that the approximate radius of the Earth is 3,963 miles.]

How do the results of the algebraic and geometric approaches differ?

[The results are typically similar, but the algebraic approach gives a formula that is an approximation, whereas the geometric approach yields an exact value.]

Why is it important to make sure that the units of measure are the same for the radius of the Earth and the height above sea level?

[The radius of the Earth is usually expressed in miles or kilometers, whereas the height above sea level is usually expressed in feet or meters. To use the Pythagorean theorem to find the hypotenuse of the triangle formed by the height and radius, all measurements must be converted to the same units.]

Extensions
 
  1. Encourage students to research this topic further. Calculating the distance to the horizon was an important question for early sailors. What other formulas were used to calculate the distance to the horizon?
  2. For students with advanced knowledge, have them consider the same situation for an ellipse. From a point on the minor axis extended, what is the distance to the horizon on the ellipse? Note that this is a very difficult question and is likely not appropriate for all students.

    The figure below shows the construction that will generate the tangent line from a point P that lies on the minor axis extended. The steps of the construction are as follows:

    1. Identify the focus F.
    2. With P as the center, construct a circle with radius PF (the blue circle in the figure below).
    3. With F as the center, construct a circle with radius equal to the major axis of the ellipse. Although this circle is not shown in the figure below, the two green segments represent radii of this circle, and this circle intersects the previous circle at Q and Q'.
    4. Segments FQ and FQ' intersect the ellipse at R and R', and these are the tangent points to the ellipse from P.

    The length PR can then be determined using the equation of the ellipse.

  3. Explore the possibility of being able to see a geographic landmark with a known altitude and distance from the top of a mountain peak.

Teacher Reflection
 
  • Did the activities in this lesson provide sufficient challenges for all students? If not, how could the activity be modified?
  • The activity sheet provides a fair level of guidance for students. How could this lesson be presented (perhaps without the activity sheet) to require more of students?
  • Were students able to effectively compare the algebraic and geometric solutions? What differences did they find when comparing the results?
  • This lesson contains several activities for students. Rate the activities in this lesson from least to most challenging.
  • How could this lesson be modified so that it could be taught without the activity sheet?

NCTM Standards and Expectations
 
Algebra 9-12
  1. Draw reasonable conclusions about a situation being modeled.
  2. Identify essential quantitative relationships in a situation and determine the class or classes of functions that might model the relationships.
Geometry 9-12
  1. Use geometric ideas to solve problems in, and gain insights into, other disciplines and other areas of interest such as art and architecture.
This lesson prepared by Samuel E. Zordak.
  
1 period   

NCTM Resources

Principles and Standards for School Mathematics

 Activities
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