# Geometry & Trigenometry problem

1. Apr 27, 2008

### someone1029

1. The problem statement, all variables and given/known data

The question sheet is the attached .jpg file

2. Relevant equations

No equations as such, however we are doing radian measure at the moment if that helps at all. Angle measurements do not have to be in radians, they can be in degrees.

3. The attempt at a solution

So far we [there are two of us trying to work on this one] have decided to use the cosine rule to find the length d, however we cannot get a value for lambda. Also we think that the dotted line in the second diagram bisects the radius that is parallel to h, we just cant prove it. If we can prove this then we can do the rest of the problem ourselves [we hope].

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2. Apr 27, 2008

### HallsofIvy

Staff Emeritus
Why would you use the cosine law? You are TOLD to use the Pythagorean theorem because this is a right triangle- the line of length d, that you are looking for, is tangent to the earth and so at right angles to a radius of the earth.

The dotted line is perpendicular to the line "r" and "h" but clearly does not bisect the radius. The greater h is the closer that will be to the center of the earth.

Last edited: Apr 27, 2008
3. Apr 27, 2008

### someone1029

Where is it stated/indicated that the line d is a tangent?
There is no right angle sign on the diagram indicating this and it isn't stated elsewhere on the sheet.

4. Apr 27, 2008

### Macka21420

hi im someone1092's partner in all this
the thing is that we cant figure out how
we cant figure out how 2 use pythagorus, because we couldnt get a full equation for d, even with subsitution and what not. We can get phi=con^-1(h/d), but we cant find an equation involing r. u would think u would use lanbda however the point at which the dotted line passes through the horizontal radius line isnt indicated, hence we cant get a ratio. note that it doesnt go through the radius line at the circumference, but at some unstated ratio.
also if it did go through the line as a tangent, then we would have cos lambda=r/r
so cos lambda=1
therefore lambda would have to equal 0, and we would have an impossible triangle.
this is why we chose to use cosine rule
you end up with d^2=r^2+(r+h)^2-2r(r+h)cos lambda
however, it is stated that we must express d in terms of r and h
hence our delma, we cant find a value for lambda

5. Apr 27, 2008

### Kurret

If d was not a tangent, then it would either be outside the earth, or it would be possible to move d so the horizon circle increases, but the horizon circle is kind of defined as being the maximum. Think about it, it is kind of obvious actually.
how did u get that?

6. Apr 27, 2008

### HallsofIvy

Staff Emeritus
The problem says that $\phi$ "is the angular separation from the horizon circle". The "horizon circle" is the what you see when your line of sight is tangent to the earth's surface.

The total of the radius of the earth, r, and the height of the satelite is the length of the hypotenuse of the right triangle. One leg is the distance d and the other is the radius of the earth r. Put that into the Pythagorean formula.