The chimney at Tiszaújváros

Adapted from IOAA 2019 (Keszthely), theory problem 6. We have restated it in our own words and kept the original data. We have left out part (b) of the original, which needs the satellite images, so parts (b) and (c) here are parts (c) and (d) of the original. The solution was last checked on 17 October 2026.

The third tallest structure in Hungary is the chimney of a power station near the town of Tiszaújváros. The two Sentinel-2 satellites photograph each area every few days, always at the same local time to within a few minutes, with a resolution of 10 m. One image of the chimney was taken on 29 June and another on 16 December, close to the two solstices. On the two images its shadow is 125 m and 780 m long.

(a) On which of the two dates do we expect the shadow to be longer? [1 point]

(b) Estimate the height of the chimney from the two lengths of its shadow. For this part only, assume that both images were taken at local noon. [16 points]

(c) Which of the following could affect the accuracy of the height found in (b)? More than one may be correct. (A) The flattened shape of Earth. (B) The limited resolution of the images and the blurred edge of the shadow. (C) The height of the base of the chimney above sea level. (D) Changes in the tilt of Earth's axis through the year. (E) Refraction in the atmosphere. [2 points]

Answer(a) The shadow is longer on 16 December. (b) The chimney is about 230 m tall. (c) The answers are (B) and (E). The chimney is in fact 250 m tall. The result is very sensitive to the angle used, since a change of 1° in ${\alpha }_1-{\alpha }_2$ changes $h$ by about 26 m. With the full 46.88° the equation has no solution at all, because these two shadows can give a difference in altitude of at most 46.37°, when $h=\sqrt{x_1{x}_2}=312$ m. The images were taken in the late morning, at about 09:45 UT, when by our calculation the Sun's altitudes were 63.1° and 18.0°. These differ by 45.0°, so the rounded value happens to suit the data.