And like usual, I've solved the problem below...

As I mentioned above, this problem can be solved using calculus or geometry, but I elected to take the calculus route. Click on any of the images below to make them larger!

Above we see my initial set up of the problem. I decided to take the first quadrant of the total image and find the shaded bit there. To do that, I had to use an entire unit circle.

### *CORRECTION:* I made a weird mistake in the very last section of my answer. I incorrectly "simplified" by adding (3/3) to ((pi)/3) and managed to get (4(pi)/3). Thank you to Brian in the comments section for pointing it out! The first part of the line, before the boxed answer, is correct.

Did you solve this a different way? Do you have any questions about my solution? Please leave a comment! I'd love to see a geometric solution in the comments section.

Hint: exploit the symmetry of the construction to reduce work.

ReplyDeleteIn the last step, how does 1 + (pi)/3 = 3/3 +(pi)/3 = (3+(pi))/3 become 4(pi)/3 all of a sudden?

ReplyDeleteThat seems bigger than the 1 square unit of the original square ( 4(Pi)/3 - (3)^(.5) = 2.45673939722)

Wow, thank you Brian. I just made a complete misstep in the easy section of the proof. I falsely attempted to simplify by adding (3/3) to ((pi)/3). I'm editing the post now to account for my mistake! The part right before the boxed answer is correct. -Tori

DeleteAlso works out quite nicely by considering (as you did) quarter parts but then use sectors from the corner of the original square and add/subtract triangles.

ReplyDelete