The problem is:
Assuming the Sun is far away from Earth and that light rays are arriving parallel to each other, determine Earth’s circumference, assuming that it is spherical.
Does anyone know how to solve this? Any hints or advice? I've been working some of the geometry but haven't gotten anything solid yet.
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Draw two right triangles, initially with corresponding legs parallel to one another. One triangle has legs of 10 m and 0.113 cm, corresponding to one pole and its shadow. The other has legs of 10 m and 1.36 m, for the other pole.
This is not correct because the hypoteneuses which correspond to the Sun's rays are not parallel. You must rotate one triangle through the difference between corresponding acute angles of the pair, to render the hypoteneuses parallel. The angle of rotation is then the amount of arc between the poles on the round Earth.
Render that arc in radians and divide into the 10 km distance between the ploes. This gives the radius of the Earth, which multiplied by $2\pi$ gives the circumference.
The diagram shows the system you have. You can work out $\theta$ and $\phi$ with standard trigonometric methods. Then $\phi-\theta$ gives you an angle, and you have the arclength of 790km. This allows you to compute the radius of the earth, and thus the circumference using $2\pi r$.