|MadSci Network: Earth Sciences|
To answer this question I had to dig up some geometry and algebra. It sounds like an easy question, but before long you have to use triangeles in circles, so I will have you set up a demonstration, and show you a picture. Since the answer depends on how far off the the earth your eyes are, I will first give you the answer as if you were standing on a 1,000 foot tower, and then if you are four feet tall. If you can work out how to do the math on your own (which I hope you try to do... or get someone to help you) then you can figure this question out for any height. The demonstration: Get an army man or some other action figure and stand the figure on a ball. You will see that the smooth surface of the ball is falling away from the action figure in all directions. That is what is happening to you as you stand in the middle of your “flat” field. Take a string and draw it from the figure’s eyes to where it skims the surface of the the ball (in math terms: becomes tangent to the sphere). This string is how far you see. To figure out this length - this is where the geometry and algebra come in. (I know you could measure the string with a ruler.... but out in the field, unless you have a very good friend, and lots of string, you just can’t use a ruler out there). So onto the MATH. Make a triangle: Start from the figures' eyes and make an imaginary line from the eyes to the very center of the ball (Line Q). From the very center of the ball, make another line to the point where the string touches the ball - is tangent to the ball (line R). The third line is the string (line S). The reason I had to involve the center of the ball is because I wanted to make the triangle a right triangle (one with a "right" or 90 degree angle) That means we can use the pythagorian theorom: a^2 + b^2 = c^2 In english this means (side a x side a) + (side b x side b) = (side c x side c). When you use this formula on a right triangle, the longest side, or hypotenuse must always be side c, so for us this is line Q. We also know another side, line R, and we can make this line a We know side R because that is the radius of the earth (3947 miles or approximately 20,840,000 feet) We know side Q because that is the radius of the earth + the height of the lookers eyes - and for this problem we will put our watcher on a 1,000 foot tower (so Q = approximately 20,841,000 feet) so a^2 = 20,840,000 ^2 = 434,305,600,000,000 and c^2 = 20,841,000 ^2 = 434,347,281,000,000 So we rearrange the original formula so we can solve for b (line S, the string) to get b^2 = c^2 - a^2 or b^2 = 434,347,281,000,000 - 434,305,600,000,000 = 41,681,000,000 next take the square root of 41,681,000,000 (which is b^2) so b = 204,159 ft or about 40 miles! Now if our person is standing on the ground and is about 4 feet hight, you can probobly guess that the distance the person would see is going to be much less than 40 miles.... in fact our 4 foot person will see about 2.4 miles I'll show the work below a (line R) = 20,840,000 feet c (line Q) = 20,840,004 feet so b^2 = 20,840,004 ^2 - 20,840,000 ^2 = 166720000 and b = 12912 feet or 2.44 miles GOOD LUCK! and many thanks for the challange Greta Hardin P.S. A few things to be aware of in problems like this! First notice that I kept using numbers with lots and lots of digits until the very end of the problem. That's because when you are doing a math problem you need to realize that all the digits are important. These are called "significant figures." In other words don't throw away too many numbers until you get to the end of the problem. If you are interested in what significant figures are, you can look them up... and this will serve you well as you go on in math and science! P.P.S. This problem can also be used using trigonometry. I'll show you the 1000 foot tower version that way. Now -to be able to find out anything about a tringle - and here we want to know the distance of S from the eyes to the tangent point - you need to know at least 3 things about the triangle.... and we do - we know 2 sides and one angle! We know side R because that is the radius of the earth ( 3947 miles or approximately 20,840,000 feet) We know side Q because that is the radius of the earth + the height of the lookers eyes - and for this problem we will put our watcher on a 1,000 foot tower (so Q = approximately 20,841,000) And we know the angle between sides S and R is 90 degrees because we made it that way on purpose. To find the length of side S we need to find out one more angle: The angle between sides S and Q will work nicely. To find angle SQ we first find its sine sin SQ = opposite side / hypotenuse = R/Q = 20,840,000/20,841,000 = 0.9995 now find the inverse of the sine and we get the angle 89.44 degrees now to get side S we can use the cosine of the angle SQ cos SQ = adjascent side/hypotenuse = S/Q cos 89.44 = S/20,841,000 ---> cos 89.44 x 20,841,000 = S S = 0.009796 x 20,841,000 = 204,159 feet or about 40 miles. See it works both ways!
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