Maths Olympiad Prep

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Geometry Difficulty 4.4 AIME Find the answer Italy

Problem:

On a sunny day a sphere is resting on horizontal ground. At a certain moment the shadow of the sphere reaches a distance of 10 meters from the point where the sphere touches the ground. At the same moment a rod of length 1 meter placed vertically on the ground casts a shadow 2 meters long. What is the radius of the sphere in meters?

Pick one

Solution

Solution:

The answer is (C)\mathbf{( C )}. The length of the rod's shadow is twice its height, so AB=12BD=5 mAB = \frac{1}{2} BD = 5~\mathrm{m}. By the Pythagorean theorem AD=55 mAD = 5 \sqrt{5}~\mathrm{m}.

Triangles ABDABD and AECAEC are similar, so AC:AD=CE:BDAC : AD = CE : BD. Setting r=CEr = CE we have: (5−r):55=r:10(5 - r) : 5 \sqrt{5} = r : 10 from which r=105−20r = 10 \sqrt{5} - 20.

Figure 1

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Source: MathNet, licensed CC-BY-4.0. Statement translated into English from it; metadata (topic, difficulty) added by this project.