Trigonemtric Functions

"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?