A Stick and A Ball
On a bright and sunny day, I took a stick (of negligible width) and a ball to an empty, perfectly flat field. I held the stick perpendicular on the field and found that its shadow was as long as the stick itself. If I placed the ball down on the field and measured its shadow, starting from the point it touches the ground, it would be times as long as the radius of the ball. Which of the following numbers is closest to the value of ?
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1 solution
The important thing here to note is that the light rays will be tangent to the sphere at some point, as shown in the cross-section above. Since the shadow of the stick is the same length as the stick, and light rays are parallel to each other, the light hits the ground at a 4 5 ∘ angle. We get ∠ A B D = ∠ A D B = ∠ D O C = 4 5 ∘ , and both A B D and O C D are isosceles right triangles. Letting O A = O C = r , we find
A B = A D = r + O D = r + O C 2 = r + r 2 = r ( 1 + 2 ) .
Thus, k = 1 + 2 , and the closest number to this is 2 . 5 .