Mt. Whitney to Death Valley

Task

The Morris family is on a road trip through California. One day they are driving from Death Valley to Sequoia National Park. Death Valley is home to the lowest point in the US at Badwater Basin with 282 feet below sea level. Sequoia National Park is home to Mt. Whitney, the highest point in the lower 48 states with 14,505 feet. Jerry is estimating from the map that the two places are only 85 miles apart as the crow flies. He is wondering:

If you hike to the top of Mt. Whitney, can you see Badwater Basin on a clear day?

Find an answer to Jerry's question and support it with an appropriate mathematical model.

(Note: In this task you may neglect the curvature of the earth and just assume that you can see long distances.)

IM Commentary

The purpose of this task is to engage students in an open-ended modeling task that uses similarity of right triangles, and also requires the use of technology (e.g., printed or electronic maps), thereby illustrating SMP 5 - Use Appropriate Tools Strategically. This task could be used as a class project or for class discussion to motivate studying similar triangles. Since students' experiences visualizing the terrain might be limited, the class could simulate the situation in the classroom: Can you see a point on the floor of the classroom when standing on a table on the other side of the room?

The bulk of the modelling occurs in the process of visualizing the top of Mt. Whitney at one end of the hypotenuse of a right triangle and Badwater Basin at the other end of the hypotenuse. The information to find the side lengths for the legs of this right triangle are given. Students will then have to consult a map or a computer program like Google Earth to investigate the terrain between Mt. Whitney and Badwater Basin. The question becomes if there are any obstacles in the line of sight, i.e. are there any mountain ranges that would block the view? Using similar triangles we find out how high those mountains at different points along the line between the two locations could be and compare them to the actual mountain heights.

An even more open version of this task would be to simply ask:

The highest and lowest points in the lower 48 states of the US are in adjacent counties in Southern California. Provided the visibility is good, can you see one from the other?

In the solution to this task we neglect the curvature of the earth and just assume that you can see long distances. A good follow-up investigation would be to find out how justified this simplification really is. An interesting and quite straight-forward geometry problem involving tangents to circles answers how far away the horizon is at different elevations above sea level. There is a discussion of this at Discover Magazine Blog. We also investigate this question in G-MD Neglecting the Curvature of the Earth.

Finally, the tasks talks about the map distance between Mt. Whitney and Badwater Basin is about 85 miles. Since the 2.75 mile height of Mt. Whitney is so small compared to this distance, the distance from the top of Mt. Whitney to Badwater Basin is only slightly larger than 85 miles ($\sqrt{85^2+2.75^2}\approx 85$). So even if students would interpret "as the crow flies" being the length of the hypotenuse of the triangle in question, it should not affect the final solution.

Solution

If we neglect the curvature of the earth, we can draw a right triangle to model this situation. The vertical leg of the triangle has length $282 \text{ ft} + 14505 \text{ ft} = 14787 \text{ ft}.$ The horizontal leg has length 85 miles or $85 \text{ mi}\cdot 5280 \text{ ft}/\text{mi}= 448800 \text{ ft}.$

We now have to determine if there are any mountain ranges between the two locations that would block the line of sight. There are several ways to find out. We can look at a topographical map to find the elevations along the direct path between the two locations. We can also use maps on the internet or computer programs like Google Earth to get this information. The picture below was generated on Google Earth. Moving the curser along the line of sight shown in the picture allows you to find the elevation at any point. We can also zoom in to find the approximately highest point in a mountain range along the line.

(Note that the elevations and distances in the picture don't agree perfectly with the ones used in the calculations. This is because the pin locations on the map were eye-balled. Fortunately the differences are small enough so that they don't matter in this case.)

We see that approximately 70 miles from Mt. Whitney or 15 miles (79200 ft) from Badwater Basin there is a mountain range (Panamint Range) with elevation above 7000 ft along our line of sight. Using similar triangles, we find the maximum height $h$ that would not block the line of sight:

$$\frac{14787}{448800} = \frac{h}{79200}, $$

or, solved for $h$,

$$h=\frac{14787\cdot 79200}{448800}= 2609.$$

Therefore, the mountain range would have to be less than 2609 ft in elevation in order to allow clear view. Since the actual elevation is above 7000 ft, the sight would be blocked.