Mathematics
Mathematics
Problem
A photograph taken at noon shows a tower and its shadow. In the photo the tower rises 12 cm, and the shadow attached to its base measures 3 cm. On the same day the actual shadow of the tower is known to be 27 m long. a) Justify that the triangle formed by the tower and its shadow in the photograph is similar to the triangle formed by the actual tower and its shadow on the ground, clearly identifying corresponding sides. b) Using the similarity, calculate the actual height of the tower. c) The photograph is enlarged so that the height of the tower image on a poster becomes 30 cm. Compute i) the new scale (number of metres of actual height represented by 1 cm on the poster), and ii) the length on the poster that will represent the tower’s shadow.Intuitive explanation
This problem hinges on similar triangles—the key insight is that the sun's angle at noon is the same everywhere, so the photo and reality create triangles with identical angles, just different sizes. Once you establish that correspondence, you can use the ratio of one pair of matching sides to find any unknown dimension. Think of it as a scaling relationship: if the photo shrinks reality by some factor, that same factor applies to every measurement.
Intuitive explanation
This problem hinges on similar triangles—the key insight is that the sun's angle at noon is the same everywhere, so the photo and reality create triangles with identical angles, just different sizes. Once you establish that correspondence, you can use the ratio of one pair of matching sides to find any unknown dimension. Think of it as a scaling relationship: if the photo shrinks reality by some factor, that same factor applies to every measurement.