Calculate Planet Radius: Trig & No Calculator

[johnsholto]

A person stands on a cliff overlooking the sea. He is 100m above the sea level and he observes the horizon to be 5mrad below the local horizontal.

How do you calculate the radius from this information without using a calculator? Trigonometry I am guessing, but I need a better hint.

 

[Simon Bridge]

Always draw the picture.

Place an origin O and draw a circle radius R about it. That's your planet.

The observer is at point A, a distance R+h from O - draw the line OA.
The point B, on the circle, is where the tangent to the circle also goes through point A.
The angle between BA and the tangent to OA (through A) is $\alpha$.

In your case, h=100m and $\alpha$=5mrad.
This will give you two right-angle triangles to work your trig on.
You may be able to make an approximation based on h<<R.

 

[johnsholto]

I tried it with:

AB^2 + R^2 = (R+100)^2

AB = (R+100)sinA

But i can't solve the equation without the calculator.

 

[HallsofIvy]

Why do you need to solve it without a calculator?

 

[johnsholto]

Punishment.

Alas, I just remembered small angle approximation.

cosA = 1-(A^2/2)

cosA = R/R+100

R ≈ 8000000

I did not enjoy this problem.

 

[Simon Bridge]

hehehe well done.
They get easier.

@hallsofivy: I suspect that it's part of the instructions in the homework - however: does not mean that a calculator cannot be used to figure out how to do it without a calculator.