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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 high-powered math. The whole solution only required geometry.

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  1. amistre64
    • 5 years ago
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    ack!!...picture would help

  2. anonymous
    • 5 years ago
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    No problem. Give me a minute.

  3. anonymous
    • 5 years ago
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    This ought to help. Essentially, finding the area of the shaded.

  4. amistre64
    • 5 years ago
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    thats easier :)

  5. amistre64
    • 5 years ago
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    we essentially got 4 areas, lets call them a,b,c, and d

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  6. anonymous
    • 5 years ago
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    When I did that, I formuated the equation a=4-b-c-d

  7. amistre64
    • 5 years ago
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    thats good... we also need to realize that: b+d = c+d

  8. amistre64
    • 5 years ago
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    b = c

  9. amistre64
    • 5 years ago
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    a = 4-2b-d right?

  10. amistre64
    • 5 years ago
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    lets 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"

  11. amistre64
    • 5 years ago
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    we have: the true area for these overlapping circles is: 2b + 2d - d right?

  12. amistre64
    • 5 years ago
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    2b + d jsut gets us right back to where we were .... can we use the circumference of a circle in our solution?

  13. amistre64
    • 5 years ago
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    2pi(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 :)

  14. amistre64
    • 5 years ago
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  15. amistre64
    • 5 years ago
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    do we know that area of an equilateral triangle?

  16. amistre64
    • 5 years ago
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    that middle triangle has an area of: sqrt(3)

  17. anonymous
    • 5 years ago
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    Thats what I did too!

  18. amistre64
    • 5 years ago
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    each side section has an area of 1 so the total unshaded area is 2sqrt(3)

  19. amistre64
    • 5 years ago
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    pi(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

  20. amistre64
    • 5 years ago
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    30 degrees is 1/6 of a circle right?

  21. amistre64
    • 5 years ago
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    so the area of each side should be 4pi/6 = 2pi/3

  22. amistre64
    • 5 years ago
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    got something turned around...hold on.

  23. amistre64
    • 5 years ago
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    360/30 = 12, so we have 1/12 of a circles area.... 4pi(1/12) = 4pi/12 = pi/3... am i right?

  24. amistre64
    • 5 years ago
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    it should be 2pi/3 for the other stuff

  25. amistre64
    • 5 years ago
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    a =4-sqrt(3)-(2pi/3) a = 12 - 3sqrt(3) - 2pi ---------------- 3 is my answer

  26. amistre64
    • 5 years ago
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    a = .1735 if I did it right :)

  27. anonymous
    • 5 years ago
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    I 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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