Question 938868: So Lost Please Help!!!
A total solar eclipse occurs when the moon passes between the earth and the sun, and the darkest shadow cast by the moon, called the umbra, hits the surface of the earth. If the umbra does not hit the surface, as shown in the following figure, then a total solar eclipse is not possible. In other words, for a total solar eclipse to occur, point P must lie inside the circle for the earth.
Assume the diameter of the sun is 870,000 miles, the diameter of the moon is 2160 miles, the diameter of the earth is 7920 miles, and the distance from the center of the sun to the center of the earth is approximately 93,000,000 miles. The distance from the moon to the earth varies, but the maximum distance from the center of the moon to the center of the earth is 252,700 miles, and is called the lunar apogee. How far is P from the center of the earth during lunar apogee? Round to the nearest thousand. Can there be a total solar eclipse during lunar apogee? Explain.
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website! 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
Notice how I made the segment OO' a red segment and I made O'P a green segment.
I've also added in segments OD and O'C shown in blue
In addition, I marked a new point R which is the intersection between the circle for the earth and the segment OO''
"the distance from the center of the sun to the center of the earth is approximately 93,000,000 miles"
So, OO'' = 93,000,000
"the diameter of the sun is 870,000 miles", so the radius is half that: 870,000/2 = 435,000
So, OD = 435,000
"the diameter of the moon is 2160 miles", so the radius of the moon is 2160/2 = 1080
So, O'C = 1080
"the diameter of the earth is 7920", so the radius of the earth is 7920/2 = 3960
So, RO'' = 3960
Using the segment addition postulate, we can say
OO'' = OR + RO''
93,000,000 = OR + 3960
93,000,000 - 3960 = OR
92,996,040 = OR
OR = 92,996,040
During lunar apogee, the moon is 252,700 miles away from the earth. This means the length of segment O'O'' is 252,700
Using the segment addition postulate, we know
OO' + O'O'' = OO''
Let y be the length of OO' and solve for y
OO' + O'O'' = OO''
y + 252,700 = 93,000,000
y = 93,000,000 - 252,700
y = 92,747,300
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.
Summary so far
OD = 435,000
O'C = 1080
RO'' = 3960
OO' = 92,747,300
OR = 92,996,040
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)
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.
From the drawing, we see that OP is the sum of OO' and O'P.
OP = OO' + O'P
OP = 92,747,300 + x
We have similar triangles, so we can set up a proportion and solve for x
(OD)/(OP) = (O'C)/(O'P)
(435,000)/(92,747,300 + x) = (1080)/(x)
435,000x = 1080(92,747,300 + x)
435,000x = 1080(92,747,300) + 1080x
435,000x = 100,167,084,000 + 1080x
435,000x - 1080x = 100,167,084,000
433,920x = 100,167,084,000
x = (100,167,084,000)/(433,920)
x = 230,842.28429
The length of O'P is approximately 230,842.28429 miles.
Use this to find OP
OP = OO' + O'P
OP = 92,747,300 + 230,842.28429
OP = 92,978,142.28429
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That's a lot of work, but we know these two important pieces of info
OP = 92,978,142.28429
OR = 92,996,040
If OP > OR, then P will be to the right of R and P will be inside the circle of the earth.
However, above we see that OP is actually less than OR. So P will actually be located to the left of point R.
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.
"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
OO'' = OP + PO''
93,000,000 = 92,978,142.28429 + PO''
93,000,000 - 92,978,142.28429 = PO''
21,857.7157099992 = PO''
PO'' = 21,857.7157099992
Question: How far is P from the center of the earth during lunar apogee?
Answer: Roughly 21,857.7157099992 miles
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Jim
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