Calculate Planet Radius: Trig & No Calculator

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Homework Help Overview

The problem involves calculating the radius of a planet based on the observation of a horizon from a height of 100m, with an angle of 5mrad below the horizontal. The context includes trigonometric principles and approximations, with an emphasis on solving without a calculator.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • Participants discuss drawing a diagram to visualize the problem, setting up a circle to represent the planet, and identifying relevant triangles. There are attempts to apply trigonometric identities and the small angle approximation, along with questions about the necessity of solving without a calculator.

Discussion Status

Some participants have offered insights into the setup of the problem and the use of approximations. There is an exploration of different approaches, but no explicit consensus has been reached on a single method or solution.

Contextual Notes

Participants note that the requirement to solve without a calculator may be part of the homework instructions, which adds a layer of complexity to the discussion.

johnsholto
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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.
 
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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.
 
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.
 
Why do you need to solve it without a calculator?
 
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.
 
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.
 

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