Stripey Circle

Geometry Level 4

Find the area of the region in the $XY$ plane consisting of all points in the set $\{(x, y)|x^{2} + y^2≤ 144 \text{ and } \sin(2x + 3y) ≤ 0\}$ .

Image Credit: Wikimedia NASA

The answer is 226.195.

1 solution

Vishnu C
May 10, 2015

Symmetry! It's really a powerful tool! The included and excluded areas are symmetrical about the line 2x+3y=0, which if you'll recall from coordinate geometry, is also a diameter of the circle. So, the area included = 72 $\pi$ =226.195.

Did the same!

Aalap Shah - 6 years, 1 month ago

Hard work pays off. Except when shortcuts help you get there faster :)

If you think that what you did was silly, checkout parth lohomi's posts. They're so tough that I avoid them almost altogether.

vishnu c - 6 years, 1 month ago

A lovely solution Vishnu! I feel a bit silly now because I used numerical integration and it took me almost an hour to figure out.

Peter Macgregor - 6 years, 1 month ago

Haha! Well, I used to dismiss symmetry as something that we humans have developed and falsely concluded that there was nothing mathematical about it. I reviewed my conclusions recently and came to realize the awesomeness of the mathematics behind it. I really want to do group theory soon. But these entrance exams kinda make me go crazy and I lose focus on what's in front of me. Mais, c'est la vie. But it'll all be over by next week:)

vishnu c - 6 years, 1 month ago

This was exactly the same solution given by CMI.

Deeparaj Bhat - 5 years, 3 months ago

Stripey Circle

Image Credit: Wikimedia NASA

The answer is 226.195.

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