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Problem of the Week
Problem D
Making an Eclipse

Quinnen made a model to demonstrate eclipses. She used a spherical LED bulb with diameter \(98\) mm to represent the Sun, and foam spheres with diameters \(30\) mm and \(14\) mm to represent the Earth and the Moon, respectively. The Earth and Sun are attached to a base using metal rods, and the Moon is connected to the Earth’s rod by a wire so that it can rotate around the Earth. The centres of the Earth, Moon, and Sun are all the same distance above the base. Quinnen rotates the Moon around the Earth and stops when the Moon is at its closest point to the Sun. In this configuration, her model demonstrates a total solar eclipse. In other words, if you were able to look towards the Sun from the point on the surface of the Earth closest to the Sun, the Moon would completely block the Sun. In this configuration, the distance between centre of the Moon and the centre of the Sun is \(360\) mm.

A side view of the model showing a circle representing the earth, a larger circle representing the sun, and a smaller circle in between representing the moon. A triangular wedge is created by two dashed lines starting from the point on the Earth closest to the Sun. One line goes diagonally up and is tangent to both the moon and the sun. The other line goes diagonally down and is tangent to both the moon and the sun.

Determine the maximum possible distance between the centre of the Earth and the centre of the Moon in Quinnen’s model.

Note: You may find the following known result about circles useful:
If a line is tangent to a circle, then the perpendicular to that line at the point of tangency passes through the centre of the circle.