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KCC's Quizzes AQQ258 about a partial eclipse Sun-Moon

1. Quote of the week: "People say nothing is impossible; but I do nothing every day!" - Anonymous

                     

2. New quiz: AQQ258 about a partial eclipse Sun-Moon

     

Some of us (in Dallas) had probably the chance to have observed a total eclipse of the Sun last week.

To celebrate that event in our quizzes, we propose the following related challenge: 

The above pictures describe a partial eclipse between the moon (circle B) and the sun (circle R).

At a certain instant, the eclipse is such that points MNO form a quarter area of circle R; meaning arc MN is a quarter circle).

At that moment, point O is the center of R and the segment OM measures 4 cm.

Question:

What is the area of the portion of blue circle B that is situated outside circle R?

Good luck! And try to be among the firsts!

Kuo-Chang

P.S. Don't hesitate to share those weekly quizzes in EZ to colleagues or friends!

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  • Let P then center of the Moon.
    P is the bisector of angle MON, by symmetry, so MOP is 45 degree.
    Triangle MOP has two sides equal, to R, radius of the Moon, so angle OMP is also 45 degree.
    So, by the triangle OPM, angle OPM is 90 degree.
    Diagonal of OMP = 4,  so by Pythagoras, R = square root of 8  ( R2+R2 = 42), or R2 = 8. 

    The sector of the Moon to the left of OM is A = ¼πR2 – ½R2
    (that is the sector radius R of 90 degree  less the area of the right triangle OPM) = 2π - 4

    The portion of the Moon hiding the Sun is the sector of radius 4 of 90 degree plus twice the previously computed Area:  As = ¼π42 +2(2π - 4) = 8π-8.
    The total area of the Moon is πR2 = 8π, thus we were hunting for an area of 8. (A little bit less than one third of the full Moon).

    My answer is 8 square cm.

  • Bingo  ! Correct answer with crystal clear development, congratulations! And you are the first!

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