# Math

Ok, I'm lost on this....

The equation D=1.2 gives the distance, D in miles that a person can see to the horizon from a height h in feet. Solve this equation for h

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1. You left out information in your equation. The equation I find for this is D=sqrt(13h). The equation in your book may be differnt.

Solve for h
D^2=13h
h=(D^2)/13

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2. A question that often arises amongst cruise passengers is how high a building can be seen from a ship at sea? A similar reverse question is how far can one see from the top of a building? An easy way to rephrase the question is to ask what is the distance from the top of a skyscraper to the horizon? Lets see if we can create a picture of the problem.

Draw yourself as large a circle as possible on a sheet of paper. Label the center O. Draw a vertical line from O to point A on the upper circumference. Extend the line past the circumference slightly to point B. Draw another line from O, upward to the right at an angle of ~30º to the vertical line, and intersecting the circumference at point C, our horizon point. Label OA and OC as r, the radius of the Earth. Label AB as h, the height of our make believe building. Label BC as d, the distance from the top of the building to the horizon or a ship at sea. Angle OCB = 90º.

From the Pythagorean Theorem, we can write that d^2 + r^2 = (r + h)^2 = r^2 + 2rh + h^2.
Simplifying, we get d^2 = 2rh + h^2 or d = sqrt[h(2r + h)].
The mean radius of the Earth is 3963 miles which is 20,924,640 feet.
Therefore, our distance d becomes d = sqrt[h(41,849,280 + h)].
If we wish to determine how far we can see from a building 1000 feet high, we need only compute d = sqrt[1000(41,849,280 + 1000)] = 204,573 feet or 38.7448 miles, ~38.74 miles.

If we were interested in determining how high a building we could see from a distance at sea, we need only solve our expression above for h which must make use of the quadratic formula. Rearranging our expression to h^2 + 2rh - d^2 = 0, we find that h = [-2r+/-sqrt(4r^2 + 4d^2)]/2 which simplifies to h = sqrt(r^2 + d^2) - r. Using our distance of ~38.75 miles calculated above, we can now solve for h = sqrt(20,924,640^2 + (38.75(5280)) - 20,924,640 which turns out to be 1000 feet.

Our expression for d can actually be simplifed somewhat due to the insignificance of h relative to r. We can easily rewrite the expression as d = sqrt(2rh) and not lose any accuracy to speak of. Taking it a step further, and since it is convenient to use h in feet, we can write d = sqrt[2(3963)miles(h)miles] = sqrt[1.5h].
...........................................................5280 ft.
Using our 1000 foot high building again with the simplified expression, we get d = sqrt[1.5(1000)] = 38.7298 miles or ~38.73 miles, or approximately 79 feet difference.

For D = sqrt(1.5h), D^2 = 1.5h making h = D^2/h, h in feet, D in miles.

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