Does Google Have an Answer for This?
The white ring, blue small circle, and tri-colored big circle are all cocentric at point O. Point A is moving around the circumference of the small blue circle and point B is moving around the circumference of the large circle at a different angular velocity. When O, A, and B are co-linear, segment
has a length of 4. When the measure of
(
is tangent to the smaller circle), segment
has a length of 8. Point P is stationary at an arbitrary position on the outside circumference of the white ring. If the minimal distance between points A and P is 1, find the area of the yellow sector.
Assume:
The yellow, blue, and green sectors are congruent.
Points P,A,B stay on their respective circumferences.
The answer is 53.41.
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1 solution
By Pythagorean theorem, we get R 2 = ( R − 4 ) 2 + 6 4 ⇒ R = 1 0 . Thus the radius of the Small circle is 6.
The white ring adds 1 unit to the radius of the blue circle, thus the area of the blue+white sectors = 4 9 π . Also, the area of the big circle = 1 0 0 π
The area of the large circle minus 4 9 π will yeild the combined areas of the green, yellow, and red sectors. Since they are congruent, the yellow region has 1/3 this area.
∴ 3 1 0 0 π − 4 9 π = 1 7 π ≈ 5 3 . 4 1