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Thread: AMC Math questions - Help?

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    AMC Math questions - Help?

    There are three questions I'm having trouble solving and would appreciate if you guys could solve and explain why (some of these are from the Year 11 and 12 section of the AMC test).

    1) Screenshot_2.jpg

    Answer is 1/4

    2) Screenshot_3.jpg

    Answer is between 7/16 and 1/2

    3) Screenshot_1.jpg

    Answer is 25pi - 25

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    Re: AMC Math questions - Help?

    1) let r be radius of cone, R be radius of quarter circle
    From quarter circle, L = r*theta, 2pi*r = R*pi/2, R = 4r
    From cone, sintheta = r/R, hence sintheta = 1/4
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    Re: AMC Math questions - Help?

    I'm gonna label the corners of the outer square with A,B,C and D to make it easier, with A being the top left corner. And the corners of the inner rectangle with E, F, G, H with E being the upper most corner. [My apologies, all these letters may seem confusing to memorise so I suggest writing them down on a diagram somewhere so the solution is a little easier to follow]

    Let angle AED = theta. From the larger unshaded triangle, we can tell that tantheta = 2.
    Since AED, HEF, and BEF are supplementary, they add to 180, and HEF equals 90 whilst AED is theta. So doing the algebra (I trust that you can do this part since this question is one where if you can see it, the maths is pretty simple), and using angle sum of triangle, EFB equals theta. Using the same proof (supplementary angles, angle sum of triangle) angle FGC is also theta. Using the tantheta identity in each of the unshaded white triangles, you can get tantheta = something. (For the upper triangle, 1/2 is already given to you as a side length.) Let the other adjacent side be x, and let
    1/2 over x = 2. (From the tantheta=2 identity you proved earlier on) And solve to get 1/4. Now since its a square the side length must be 1, so 1-1/4 gives 3/4 as the opposite angle in tantheta=2 for the bottom one.

    (Oh, just a quick reminder that tantheta = opposite/adjacent, so the fraction itself can equal 2. Don't put tan(fraction)=2!)

    Repeat the process for the bottom one, and you should get 3/8. Then, using pythagoreas theorem in each of the triangles, you should get respectively, rt(5)/4 and 3rt(5)/8. (Remember to square root the squares! It's a common silly error :P)

    Then multiply them. I got 15/32 which is between 7/16(7/16=14/32) and 1/2. (1/2=16/32)
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    Re: AMC Math questions - Help?

    Quote Originally Posted by _Anonymous View Post
    There are three questions I'm having trouble solving and would appreciate if you guys could solve and explain why (some of these are from the Year 11 and 12 section of the AMC test).

    1) Screenshot_2.jpg

    Answer is 1/4

    2) Screenshot_3.jpg

    Answer is between 7/16 and 1/2

    3) Screenshot_1.jpg

    Answer is 25pi - 25

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    Re: AMC Math questions - Help?

    Quote Originally Posted by 1729 View Post
    yeah I got 50 as well

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    Re: AMC Math questions - Help?

    Quote Originally Posted by 30june2016 View Post
    yeah I got 50 as well
    Last edited by 1729; 19 Jul 2017 at 6:20 PM.
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    Re: AMC Math questions - Help?

    Quote Originally Posted by 1729 View Post
    Ah, wow I did it differently and never thought of something as fast and simple as that.

    But anyway, the much longer way I thought of for the question was:

    I drew a 10x10cm square around the circle, and drew circles of 5cm radii as shown.




    From there I just subtracted the green area ([square - circle]/2)

    Then subtracted the 2 quarter circles (the navy) and got an answer of 50 HAHAHAHAHAHAHAHAH way more trek though
    Last edited by 30june2016; 19 Jul 2017 at 6:37 PM.
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    Re: AMC Math questions - Help?

    Yep, the answer is 50.

    25pi/4+25pi/4+(25-25pi/4)+(25-25pi/4)

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    Re: AMC Math questions - Help?

    Yep my bad, the answer was 50.

    Thank you all for your help.

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    Re: AMC Math questions - Help?

    Quote Originally Posted by 1729 View Post
    I sort of understand your method, but am having trouble visualising it. Is there any way you can draw it or show a diagram?

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    Re: AMC Math questions - Help?

    Quote Originally Posted by Mongose528 View Post
    Yep, the answer is 50.

    25pi/4+25pi/4+(25-25pi/4)+(25-25pi/4)
    How did you get those numbers?

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    Re: AMC Math questions - Help?

    The radius of the circle is 5, so the area is 25pi

    Area of shape = Area of Quarter circle + Area of Quarter circle + (25 - Area of Quarter circle) + (25 - Area of Quarter circle)

    25 is the area of a 5*5 square

    The top left and bottom right form quarter circles

    the top right and bottom left areas are given by 25 - Area of Quarter circle

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    Re: AMC Math questions - Help?

    Quote Originally Posted by _Anonymous View Post
    I sort of understand your method, but am having trouble visualising it. Is there any way you can draw it or show a diagram?
    Draw two adjacent squares ABCD and BCEF sharing a common side BC. This forms a rectangle with a ratio 1:2. Then draw a semi-circle passing through D, B and E with diameter DE and centre at C. The arc of this semi-circle and the side BC divides the rectangle into four parts - two congruent quarter circles and two other congruent figures. Rearrange these four parts into the figure given in the question and you will notice that it can be made. Then reverse the process to understand how the rectangle is formed.

    Otherwise you can draw two perpendicular lines connecting the opposite corners in the figure given in the question and you will notice that it divides the figure into the same four parts made by the rectangle division. It is just a matter of visualisation, rotation and translation.

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    Re: AMC Math questions - Help?

    Quote Originally Posted by 30june2016 View Post
    yeah I got 50 as well
    same, i was just afraid to say it cauz...lol

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