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You are an astronomer on planet Nearth, which orbits a distant star. It has recently been accepted that Nearth is spherical in shape, though no one knows its size. One day, while studying in the library of Alectown, you learn that on the equinox your sun is directly overhead in the city of Nyene, located 1600 kilometers due north of you. On the equinox, you go outside and observe that the altitude of your sun is 80 ∘.What is the circumference of Nearth?

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cryssatemp

Answer: 7200 km

Explanation:

In this case of Nearth, we will have to do something similar to what Eratosthenes did to calculate the circumference of Earth:

We have two places separated by a distance of [tex]1600 km[/tex]:

-In the first place, Nyene, the Sun is directly overhead. This means that if we put a stick in this place we will not see shadow.

-In the second place, when we put a stick, we will observe a shadow of [tex]80\°[/tex]

Taking into account the sun is radiating to both places at the same time, but in the first place there is no shadow and in the second place we observe a shadow, the logical explanation is that this must be a curved surface (this is what Eratosthenes concluded on Earth and is the same on Nearth).

So, by trigonometry we know the angle the sun makes with the second stick is the same angle of an arc of the spherical planet. If we know this value, we can find the whole circumference (perimeter) of this sphere by the Rule of Three and taking into account a circumference has  [tex]360\°[/tex]:

If [tex]80\°[/tex] ----- [tex]1600 km[/tex]

  [tex]360\°[/tex] -----  [tex]?[/tex]

Then:

[tex]?=\frac{(360\°)(1600 km)}{80\°}[/tex]

[tex]?=7200 km[/tex] This is the circumference of Nearth

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