Question 126730: From a space probe circling Io, one of Jupiter's moons, at an altitude of 552km, it was observed that the angle of depression at the horizon was 39.7 degrees. What is the radius of Io?
Answer by Earlsdon(6294)   (Show Source):
You can put this solution on YOUR website! Well, it is said that "a picture is worth a thousand words"
In this problem, a picture would be very useful but I don't know how to do one, so I'll use words (not a thousand of 'em, hpoefully).
1) First, draw a circle that represents Io, the moon.
2) Next, place a point outside of this circle, and this represents the orbiting space probe.
3) Now draw a straight line connecting the space probe to the center of the circle (Io). The distance represented by the length of this line is the sum of the altitude of the space probe (552km) and the radius (R) of Io which is what we are trying to find.
4) Now draw a straight line from the space probe to the edge of the circle (Io) so that it is tangent to the circle.
5) The angle of depression is given as 39.7 degrees and this is the angle between the two lines you have drawn. Correction! The angle between the two lines you have drawn is 90-39.7 or 50.3 degrees.
6) Draw one more line from the center of the circle to the point of tangency of the second line. This is the radius of IO and can be labeled R.
The angle between these two lines is, of course, 90 degrees.
You now should see a right triangle whose hypotenuse is the first line you drew and whose length is 552+R.
The base of this right triangle is just R, the radius.
Now we can apply a little trigonometry and algebra to find R, the radius of IO.
The sine of the angle of depression (39.7 degrees) is defined to be the side opposite the angle (the base of our right triangle which is just R) over the hypotenus of the triangle which is 552+R, so we can write:
 Multiply both sides by (552+R).
 Simplify.
 Subtract 0.6388R from both sides.
 Finally, divide both sides by 0.3612
So, the radius of Io is 976.2 km.
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Correction!
In solving this problem, I mistakenly used the wrong angle.
The angle of depression is given as 39.7 degrees but to solve the problem, I should have used the sine of (90-39.7)(Sin(50.3) rather than the sine of 39.7.
This changes the answer as follows:
The corrected answer is:
The radius of IO is 1847.8 Km
My sincere appologies for this error.
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