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You are standing on the top of a mountain, looking out towards the horizon. From your lofty vantage point, you can't help but wonder, "How far is it to the horizon anyway?"
If your eye is h miles above the Earth, then you can see a distance d of miles to the horizon. Your line of sight to the horizon forms a right angle with a radius from the center of the Earth, as shown. The radius of the Earth is approximately 4,000 miles.
(visual in the attached image)
This visual model is intentionally skewed to help you visualize the right triangle. At the top of the hemisphere, a mountain is shown with the distance from the surface of the hemisphere to the top of the mountain indicated as the height of the mountain. A dashed diagonal line, labeled d, extends from the top of the mountain to a point on the hemisphere. The two ends of the dashed line are connected to the center of the hemisphere by solid lines. The two solid lines and the dashed line form a right triangle. A right angle symbol is shown where the dashed line meets the surface of the hemisphere and connects with a radius of the hemisphere. The distance from the center of the hemisphere to the surface is labeled 4,000 miles.
If you were in the northern Pakistan and standing on top of Broad Peak, which is about 4 miles tall, you would be able to see ____ miles to the horizon.
If you were in eastern France and standing on top of Mont Blanc, which is 3 miles tall, you would only be able to see ___ miles to the horizon.
