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By measuring the time lapse between two sunsets from a beach, we can estimate the radius of the Earth. This can be done using the concept of basic trigonometry.
When the Sun is setting, it appears to move along the horizon. The time it takes for the Sun to complete one full circle (24 hours) can be divided into 360 degrees. Therefore, every hour the Sun moves approximately 15 degrees.
By measuring the time it takes for the Sun to set completely (e.g. from when it touches the horizon to when it disappears), we can estimate the angle between our location on the beach and the center of the Earth. This angle is related to the radius of the Earth. With some simple calculations, we can use this angle to estimate the radius of the Earth.
Hence, option 3, the radius of the Earth, is the correct estimation that can be made by measuring the time lapse between two sunsets.