Identifying Mountain Peaks in Hawaii.

You are standing on Oahu and you can see three mountain peaks in the horizon. The possible mountains and their peaks are: Lanaihale which is 3,370ft; Haleakale at 10,023ft; Mauna Kea at 13,796; and lastly Kamakow at 4,961ft. Lanaihale is on Lanai which is 65 miles from you, Haleakale is on Maui and 10 time miles away, Mauna Kea is on Hawaii which is 190 miles from you and Kamakow is on Molokai which is 40 miles away.

a) Assume the Earth is 3960 miles, detirmine the angle formed at the center of the Earth.

b) Determine the Hypotenuse, which mountain is not visible form Oahu.

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Originally Posted by jameseut

You are standing on Oahu and you can see three mountain peaks in the horizon. The possible mountains and their peaks are: Lanaihale which is 3,370ft; Haleakale at 10,023ft; Mauna Kea at 13,796; and lastly Kamakow at 4,961ft. Lanaihale is on Lanai which is 65 miles from you, Haleakale is on Maui and 10 time ?? miles away, Mauna Kea is on Hawaii which is 190 miles from you and Kamakow is on Molokai which is 40 miles away.

a) Assume the Earth is 3960 miles, detirmine the angle formed at the center of the Earth.

b) Determine the Hypotenuse, which mountain is not visible form Oahu.

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in radians, $\displaystyle \theta = \frac{s}{R}$

consider a mountain $\displaystyle s$ miles from your position (assumed at sea level) whose peak is just even with your horizontal line of sight as shown in the diagram.

$\displaystyle \cos{\theta} = \frac{R}{R+h}$

$\displaystyle R+h = \frac{R}{\cos{\theta}}$

$\displaystyle h = \frac{R}{\cos{\theta}} - R$

note that $\displaystyle h$ will be in miles.