Identifying the Mountains of the Hawaiian Islands seen from Oahu"

A wealthy tourist with a strong desire for the perfect scrapbook has called in your consulting firm for help in labeling a photograph he took on a clear day from the southeast shore of Oahu.  He shows you the photograph in which you see mostly sea and sky.  But on the horizon are three mountain peaks, equally spaced and apparently all of the same height.  The tourist tells you that the T-shirt vendor at the beach informed him that the three mountain peaks are the volcanoes of Lanai, Maui, and the Big Island (Hawaii).  The vendor added, "You almost never see the Big Island from here."

The tourist thinks his photograph may have some special value.  He has some misgivings, however.  He is wondering if, in fact, it is ever really possible to see the Big Island from Oahu.  He wants accurate labels on his photos.  So he asks you for help.

You realize that some trigonometry is needed as well as a good atlas and encyclopedia.  After doing some research you discover the following facts:

The heights of the peaks vary.  Lanaihale, the tallest peak on Lanai, is only about 3370 feet above sea level.  Maui's Haleakala is 10,023 feet above sea level.  Mauna Kea on the Big Island is the highest of all, 13,796 feet above sea level.  (If measured from its base on the ocean floor, it would qualify as the tallest mountain on Earth.)

Measuring from the southeast corner of Oahu, the distance to Lanai is about 65 miles, to Maui it's about 110 miles, and to the Big Island it is about 190 miles.  These distances, measured along the surface of the Earth, represent the length of the arc that originates at sea level on Oahu and terminates in an imaginary location directly below the peak of each mountain at what would be sea level.

1. These volcanic peaks are all different heights.  If your wealthy client took a photo of three mountain peaks that all look the same height, how could they possibly represent these three volcanoes?
   Solution: The reason why they all look the same heights is because they are at differences distances from the coast.
2. The radius of the earth is approximately 3960 miles.  Based on that figure, what is the circumference of the earth?  How is the distance between islands related to the circumference of the earth?
3. To determine which of these mountain peaks would actually be visible from Oahu, you need to consider that the tourist is standing on the shore and looking "straight out" would have a line of sight tangent to the surface of the earth at that point.  Make a rough sketch of the right triangle formed by the tourist's line of sight, the radius from the center of the earth to the tourist, and a line from the center of the earth that passes through Mauna Kea.  Can you determine the angle formed at the center of the earth?
4. What would the length of the hypotenuse of the triangle be?  Can you tell from that whether or not Mauna Kea would be visible from Oahu?
5. Repeat this procedure with the information about Maui and Lanai.  Will they be visible from Oahu?
6. There is another island off Oahu that the T-shirt vendor did not mention.  Molokai is about 40 miles from Oahu, and it's highest peak, "Kamakou," is 4961 feet above sea level.  Would Kamakou be visible from Oahu?
7. You should now be prepared to name the three volcanic peaks in the photograph.  How do you decide which one is which?

Price: $15.03

Solution: The downloadable solution consists of 6 pages, 903 words and 2 charts.
Deliverable: Word Document