Question

Scientists can estimate the depth of craters on the moon by studying the lengths of their shadows in the craters. Find the depth of a crater, if the shadow is estimated to be 400 m long and the angle of elevation of the sun is 48°.

Answer 1

It's a much more interesting problem when we have to estimate the shadow length from a photo.

The crater is a curvy bumpy surface, but let's approximate it as part of a sphere and just conflate the arc and chord length, so the 400 m shadow corresponds to a 400 m chord from the edge at the surface to the bottom.

The shadow as a chord forms a hypotenuse right triangle with the depth d being the leg opposite the 48 degree angle of the sun.

$d = 400 \sin 48 \approx 297 \textrm{ m}$

Answer: 297 m