A cone with radius 6 and height 8 is sitting on a table. A spotlight with a cylindrical beam of radius 3 is shining directly at the cone. The beam is parallel to the surface of the table and the bottom of the beam just touches the table. The cone just barely blocks the entire light beam.
What is the illuminated area?
Steven's answer 42.130 in his page of math is greater than my 33.705 by a factor of almost exactly 5/4 (note: on his post he wrote 42.130)
His fancy calculus is far beyond what I remember how to read, so I can't judge it. I just mapped the circle to different views. The top view seemed easier based on the net view so I went with it. Then I integrated by arcs. Of course the arcs in the top view correspond to thinker arcs in the net. I think what I did was introduce a correction factor of 4/3 instead of 5/3 (scale r=6 to r=10).
In fact if you look at my Desmos drawing
you can estimate the relative areas by counting the squares. I get about 25.5 in the top view and 42.5 in the net view. This agrees very well. (25.5*4/3=34 which I calculated.)
Edited on November 12, 2021, 5:43 pm
Posted by Jer
on 2021-11-12 17:37:46