Distance to Horizon: confirm officer's approximation

[soccerisgreat]

On the bridge of a ship at sea, the captain asked the new young officer standing next to him to determine the distance to the horizon. The officer took a pencil and paper and in a few moments cam up with an answer. d=5/4sqrt. h

Show that this is a good approximation of the distance in miles to the horizon if h is the height in feet of the observer above the water. Assume the radius of the earth is 4000 miles. If the bridge was 88 feet above the water, what was the distance to the horizon?

I must use the power of a point to accomplish this. I really need help with this one. Thanks.

 

[Deleted member 4993]

Re: Distance to Horizon

This is a simple application of definition of horizon and Pythagorus,

Start with drawing a sketch. Then show/explain your work/thoughts and exactly where you are stuck.

 

[galactus]

Re: Distance to Horizon

Here's a diagram. As SK said, use Pythagoras to derive the general formula. Remember, it's an approximation.

Also, remember that h is in feet and is small by comparison to the radius of the Earth.

\(\displaystyle (4000+h)^{2}=D^{2}+4000^{2}\)

 

[Dr. Flim-Flam]

A cute one. Since d is in miles and h is in feet,we must have same units to solve.

Ergo (4000 + h)^2 = d^2 +(4000)^2; d^2 = 4000^2 -4000^2 +8000h +h^2.

d= sqrt[8000miles*hfeet +h^2feet] = sqrt[(8000h+h^2)/5280] miles. We can disregard h^2, since h in feet is so small.

Hence d = about sqrt[8000h/5280] miles = about (5/4)/sqrt(h), h in feet and d in miles.

Check: Assume you are 80 feet above the water.

Then d = sqrt[(8000*80 + 80^2)/5280] = about 11.06 miles.d = sqrt[(8000*80)/5280] = about 11.0096 miles

and d = (5/4)*sqrt(80) = about 11.18 miles. Using the formula will always be a little higher estimate, but good for

government work. I think Captain Blythe pulled this one on Christian.