Algebra.Com's Answer #572050 by jim_thompson5910(35256)![\"\"](\"/images/stars/stars-13.gif\")  
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First draw out the picture without the fancy visuals to distract you. So just draw out 3 circles along with tangents for the first two circles like so\r
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document.write( "Notice how I made the segment OO' a red segment and I made O'P a green segment.
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document.write( "I've also added in segments OD and O'C shown in blue
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document.write( "In addition, I marked a new point R which is the intersection between the circle for the earth and the segment OO''\r
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document.write( "\"the distance from the center of the sun to the center of the earth is approximately 93,000,000 miles\"
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document.write( "So, OO'' = 93,000,000\r
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document.write( "\"the diameter of the sun is 870,000 miles\", so the radius is half that: 870,000/2 = 435,000
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document.write( "So, OD = 435,000\r
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document.write( "\"the diameter of the moon is 2160 miles\", so the radius of the moon is 2160/2 = 1080
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document.write( "So, O'C = 1080\r
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document.write( "\"the diameter of the earth is 7920\", so the radius of the earth is 7920/2 = 3960
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document.write( "So, RO'' = 3960
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document.write( "Using the segment addition postulate, we can say\r
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document.write( "OO'' = OR + RO''
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document.write( "93,000,000 = OR + 3960
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document.write( "93,000,000 - 3960 = OR
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document.write( "92,996,040 = OR
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document.write( "OR = 92,996,040\r
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document.write( "During lunar apogee, the moon is 252,700 miles away from the earth. This means the length of segment O'O'' is 252,700
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document.write( "Using the segment addition postulate, we know
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document.write( "OO' + O'O'' = OO''
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document.write( "Let y be the length of OO' and solve for y\r
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document.write( "OO' + O'O'' = OO''
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document.write( "y + 252,700 = 93,000,000
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document.write( "y = 93,000,000 - 252,700
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document.write( "y = 92,747,300\r
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document.write( "Therefore, the distance from the center of the sun to the center of the moon during a lunar eclipse and during a lunar apogee is roughly 92,747,300 miles.\r
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document.write( "Summary so far
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document.write( "OD = 435,000
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document.write( "O'C = 1080
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document.write( "RO'' = 3960
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document.write( "OO' = 92,747,300
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document.write( "OR = 92,996,040\r
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document.write( "Let's go back to the drawing. Focus on just triangle ODP and the smaller triangle O'CP and add in the measurements you see in the summary above (well with the exception of RO'' and OR)\r
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document.write( "![](\"http://i150.photobucket.com/albums/s91/jim_thompson5910/k370ln99s2m83265af5r-2_zps115a8004.png\") \r
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document.write( "Now let x be the length of O'P shown in green. We need to find the length of x to help us solve this problem.\r
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document.write( "From the drawing, we see that OP is the sum of OO' and O'P.
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document.write( "OP = OO' + O'P
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document.write( "OP = 92,747,300 + x\r
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document.write( "We have similar triangles, so we can set up a proportion and solve for x\r
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document.write( "(OD)/(OP) = (O'C)/(O'P)
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document.write( "(435,000)/(92,747,300 + x) = (1080)/(x)
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document.write( "435,000x = 1080(92,747,300 + x)
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document.write( "435,000x = 1080(92,747,300) + 1080x
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document.write( "435,000x = 100,167,084,000 + 1080x
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document.write( "435,000x - 1080x = 100,167,084,000
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document.write( "433,920x = 100,167,084,000
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document.write( "x = (100,167,084,000)/(433,920)
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document.write( "x = 230,842.28429\r
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document.write( "The length of O'P is approximately 230,842.28429 miles.
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document.write( "Use this to find OP\r
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document.write( "OP = OO' + O'P
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document.write( "OP = 92,747,300 + 230,842.28429
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document.write( "OP = 92,978,142.28429\r
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document.write( "That's a lot of work, but we know these two important pieces of info\r
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document.write( "OP = 92,978,142.28429
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document.write( "OR = 92,996,040\r
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document.write( "If OP > OR, then P will be to the right of R and P will be inside the circle of the earth.
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document.write( "However, above we see that OP is actually less than OR. So P will actually be located to the left of point R.\r
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document.write( "Therefore, the eclipse is NOT possible. No shadow from the moon is cast at all. The moon is simply too far away from the earth.\r
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document.write( "\"How far is P from the center of the earth during lunar apogee?\" It's asking for the length of PO'', so let's find that\r
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document.write( "OO'' = OP + PO''
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document.write( "93,000,000 = 92,978,142.28429 + PO''
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document.write( "93,000,000 - 92,978,142.28429 = PO''
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document.write( "21,857.7157099992 = PO''
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document.write( "PO'' = 21,857.7157099992\r
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document.write( "Question: How far is P from the center of the earth during lunar apogee?
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document.write( "Answer: Roughly 21,857.7157099992 miles\r
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document.write( "If you need more one-on-one help, email me at jim_thompson5910@hotmail.com. You can ask me a few more questions for free, but afterwards, I would charge you ($2 a problem to have steps shown or $1 a problem for answer only).\r
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document.write( "Jim\r
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