How Do You Calculate the Radius of Planet Oceania?

[mnmakrets]

An astronaut arrives on the planet Oceania and climbs to the top of a cliff overlooking the sea. The astronaut’s eye is 100 m above the sea level and he observes that the horizon in all directions appears to be at angle of 5 mrad below the local horizontal.

What is the radius of the planet Oceania at sea level?

How far away is the horizon from the astronaut?

[Hint: the line of sight from the astronaut to the horizon is tangential to surface of the planet at sea level.]


Having drawn out the problem I have two equations:
Pythagoras: x^2 + R^2 = (R+100)^2
Sine Law: x= (R+100)sinA

x is the distance between the horizon and the astronaut
R is the radius of the planet
A is the angle between the two lengths of radius of the planet (sorry might sound a bit confusing, it makes more sense if you draw it out!)

I then tried to combine them and solve for the radius of the planet R but I am not having any luck getting the right answer :(

Any steps, answer, and/or hints are all greatly appreciated! :)

 

[Simon Bridge]

Welcome to PF;

It helps to be careful about what you call things...

You should have drawn a triangle OAB.
O is at the center of the planet.
A is where the astronaut is standing,
B is on the surface of Oceania at sea level.

Angles:
The angle OBA is $\frac{\pi}{2}$ (working in radians).
The angle AOB is $\alpha$
The angle OAB is $\frac{\pi}{2}-\alpha$

Lengths of the sides:
$|OB|=R$ (radius of Oceania at sea level)
$|OA|=R+a$ (a=100m) it is best to keep variables.
$|AB|=x = R\tan\alpha$

You will also need another triangle: ABC
points A and B are as before.
point C is on the same line as OB, further out from Oceania.
The line AC is perpendicular to the line OA.
The angle CAB is given to you - from the drawing you should be able to see how it is related to $\alpha$.

You now have three right-angled triangles.
... but I'm afraid you'll probably have to attach a diagram.

 

[mnmakrets]

Thank you, I followed the steps you have given me and I drew the first triangle, however I am having trouble understanding where the line AC should go, I made the AC line red in the diagram, I am not sure if I drew it correctly. I attached the diagram which I drew in a word doc., I don't know any other way of drawing diagrams online, I hope its not an inconvenience.

 

[Simon Bridge]

The line AC is supposed to be the local horizontal to A (the Astronaut).
The line AB is the line to the horizon as seen from A.

I attached the diagram which I drew in a word doc., I don't know any other way of drawing diagrams online, I hope its not an inconvenience.

It is poor netiquette to post attachments in poorly documented formats like docx - or any MS Office format.
Not everyone can read them.

For diagrams, please use png or gif format.
For photographs: jpg is usually good enough.
For general documents, please use an open document format or pdf

The attachment function of PF let's you upload from your computer - another approach is to use a service like google-docs.

 

[Chestermiller]

From your figure, R/(R+a) = cosα. You know α, so you can calculate R.