Height of a hill
- Round
- Theory
- Points
- 10
- Level
- Bronze
- Taught in
- Section 1.3
Adapted from IOAA 2018 (Beijing), theory problem 4. We have restated it in our own words and kept the original data, adding a value for Earth's radius. The solution was last checked on 15 October 2026.
Two friends want to measure the height of a hill next to their village, at latitude 40°. One of them climbs to the top of the hill and agrees to send a light signal to the other, in the village, the moment she sees the Sun set. On 21 March, when they do the experiment, the friend in the village receives the signal 4.1 minutes after the Sun has set as seen from the village. Estimate the height of the hill, and the distance to the horizon for the friend on the hilltop. Ignore atmospheric refraction, and take Earth's radius to be 6378 km.
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On 21 March the Sun is on the celestial equator, so it sets due west, moving along the equator. At latitude 40° the celestial equator meets the horizon at
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The Sun turns with the sky at about 15° an hour, or 15′ a minute. That is a little slower than the stars' 15.04° an hour, because the Sun moves eastward among the stars by about a degree a day. In 4.1 minutes it moves
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The friend on the hilltop sees the Sun set when it is
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By (1.6) the horizon is
AnswerThe hill is about 600 m high, and the horizon is about 87 km from the top. The official solution lets the Sun move at the stars' rate, 15.04′ a minute, and gets 602 m and 87.6 km by the same method.
This problem also appears among the olympiad problems of Section 1.3, Right ascension, declination and hour angle, which teaches the ideas it uses. Equation and figure numbers refer to the course.