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anonymous
 5 years ago
Given:
Square ABCD with side length 2.
Two quarter circles, one with center A and the other at center B. Call their intersection E. What is the area enclosed by Arc CE, Arc DE, and side CD?
I've already done the problem, but I want to verify my answer.
Also, my solution does not use calculus, trigonometry, and any other form of highpowered math. The whole solution only required geometry.
anonymous
 5 years ago
Given: Square ABCD with side length 2. Two quarter circles, one with center A and the other at center B. Call their intersection E. What is the area enclosed by Arc CE, Arc DE, and side CD? I've already done the problem, but I want to verify my answer. Also, my solution does not use calculus, trigonometry, and any other form of highpowered math. The whole solution only required geometry.

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amistre64
 5 years ago
Best ResponseYou've already chosen the best response.0ack!!...picture would help

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0No problem. Give me a minute.

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0This ought to help. Essentially, finding the area of the shaded.

amistre64
 5 years ago
Best ResponseYou've already chosen the best response.0we essentially got 4 areas, lets call them a,b,c, and d

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0When I did that, I formuated the equation a=4bcd

amistre64
 5 years ago
Best ResponseYou've already chosen the best response.0thats good... we also need to realize that: b+d = c+d

amistre64
 5 years ago
Best ResponseYou've already chosen the best response.0lets try to fill in some values here: lets say b = 5, c = 5, and d = 10 we know that: (5+10) + (5+10) but the area here is exagerated by "an extra 10"

amistre64
 5 years ago
Best ResponseYou've already chosen the best response.0we have: the true area for these overlapping circles is: 2b + 2d  d right?

amistre64
 5 years ago
Best ResponseYou've already chosen the best response.02b + d jsut gets us right back to where we were .... can we use the circumference of a circle in our solution?

amistre64
 5 years ago
Best ResponseYou've already chosen the best response.02pi(2) = 4pi for a whole circle; which means that pi is the length for a quarter of it......I think I got it. unless you already know the answer :)

amistre64
 5 years ago
Best ResponseYou've already chosen the best response.0do we know that area of an equilateral triangle?

amistre64
 5 years ago
Best ResponseYou've already chosen the best response.0that middle triangle has an area of: sqrt(3)

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0Thats what I did too!

amistre64
 5 years ago
Best ResponseYou've already chosen the best response.0each side section has an area of 1 so the total unshaded area is 2sqrt(3)

amistre64
 5 years ago
Best ResponseYou've already chosen the best response.0pi(r^2) is the area of a complete circle the r here =2 4pi is the total area of our representative circle... we only want 30 out of 360 of that circle

amistre64
 5 years ago
Best ResponseYou've already chosen the best response.030 degrees is 1/6 of a circle right?

amistre64
 5 years ago
Best ResponseYou've already chosen the best response.0so the area of each side should be 4pi/6 = 2pi/3

amistre64
 5 years ago
Best ResponseYou've already chosen the best response.0got something turned around...hold on.

amistre64
 5 years ago
Best ResponseYou've already chosen the best response.0360/30 = 12, so we have 1/12 of a circles area.... 4pi(1/12) = 4pi/12 = pi/3... am i right?

amistre64
 5 years ago
Best ResponseYou've already chosen the best response.0it should be 2pi/3 for the other stuff

amistre64
 5 years ago
Best ResponseYou've already chosen the best response.0a =4sqrt(3)(2pi/3) a = 12  3sqrt(3)  2pi  3 is my answer

amistre64
 5 years ago
Best ResponseYou've already chosen the best response.0a = .1735 if I did it right :)

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0I didn't evaluate for that, but the formula for a is what I got too! Awesome, thanks! You just earned a fan.
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