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Math Help - Geo Circle Formulas Helps

  1. #1
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    Geo Circle Formulas Helps

    I have a pizza. The radius is 10 inches long. The pizza was cut into 16 equal slices. When 1 slice was left, my sister and I both wanted it, so we agreed to cut it in half, but I like the crust more than she does, so we decided to cut it the "other way." In other words, the two pieces would not be symmetrical. The inside piece would contain all topping, and the outer piece would contain some topping and some crust.
    1. Find the area of the whole pizza.10 x 10 = 100

    2. What is the area of one piece of pizza?100/16=6.25^2=39.0625
    3. What is the area of a half-piece?39.0625/2=19.53125

    4. What would the area of the whole pizza be if it were made of half pieces?19.53125^2= 95.367431640625

    5. What is the radius of a half-piece? (ie, where do I need to cut to make two equal halves out of a piece?)9.765625

    If I am doing this this wrong it would be useful to attain the formulas needed to complete these problems. Any help much appreciated.
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  2. #2
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    Quote Originally Posted by KevinVM20 View Post
    I have a pizza. The radius is 10 inches long. The pizza was cut into 16 equal slices. When 1 slice was left, my sister and I both wanted it, so we agreed to cut it in half, but I like the crust more than she does, so we decided to cut it the "other way." In other words, the two pieces would not be symmetrical. The inside piece would contain all topping, and the outer piece would contain some topping and some crust.
    1. Find the area of the whole pizza.10 x 10 = 100
    Is the pizza a square or a circle? You said the pizza's radius is 10 inches long, so that implies a circle. But then you wrote "10 x 10 = 100," which implies a square.

    You probably meant circle, so
    A = \pi r^2 = \pi \cdot 10^2 = 100\pi

    This will affect your subsequent answers, of course, so try again!


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  3. #3
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    I understand that part. Are the rest of my calculations correct?

    Quote Originally Posted by yeongil View Post
    Is the pizza a square or a circle? You said the pizza's radius is 10 inches long, so that implies a circle. But then you wrote "10 x 10 = 100," which implies a square.

    You probably meant circle, so
    A = \pi r^2 = \pi \cdot 10^2 = 100\pi

    This will affect your subsequent answers, of course, so try again!


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  4. #4
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    Quote Originally Posted by KevinVM20 View Post
    I have a pizza. The radius is 10 inches long. The pizza was cut into 16 equal slices. When 1 slice was left, my sister and I both wanted it, so we agreed to cut it in half, but I like the crust more than she does, so we decided to cut it the "other way." In other words, the two pieces would not be symmetrical. The inside piece would contain all topping, and the outer piece would contain some topping and some crust.
    1. Find the area of the whole pizza.10 x 10 = 100
    See yeongil's post
    2. What is the area of one piece of pizza?
    A_{piece}=\dfrac1{16} \cdot \pi \cdot r^2
    3. What is the area of a half-piece?
    A_{\frac12 p}=\dfrac12 \cdot \dfrac1{16} \cdot \pi \cdot r^2
    4. What would the area of the whole pizza be if it were made of half pieces?
    I don't understand this question, sorry!
    5. What is the radius of a half-piece? (ie, where do I need to cut to make two equal halves out of a piece?)
    I assume that the central angle of the half-piece (that's the angle of the apex) is the same as the angle of a complete piece. Let denote \rho the radius of the half-piece:

    A_{\frac12 p}=\dfrac1{16} \cdot \pi \cdot  \underbrace{\dfrac12 \cdot r^2}_{this\ is\  \rho^2}

    \rho^2=\dfrac12 r^2~\implies~ \rho=\dfrac r2 \cdot \sqrt{2}

    In your case \rho \approx 7.07

    If I am doing this this wrong it would be useful to attain the formulas needed to complete these problems. Any help much appreciated.
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  5. #5
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    It is all starting to click. However, I do not see how you got the 7.07.

    Quote Originally Posted by earboth View Post
    See yeongil's post


    A_{piece}=\dfrac1{16} \cdot \pi \cdot r^2

    A_{\frac12 p}=\dfrac12 \cdot \dfrac1{16} \cdot \pi \cdot r^2
    I don't understand this question, sorry!

    I assume that the central angle of the half-piece (that's the angle of the apex) is the same as the angle of a complete piece. Let denote \rho the radius of the half-piece:

    A_{\frac12 p}=\dfrac1{16} \cdot \pi \cdot  \underbrace{\dfrac12 \cdot r^2}_{this\ is\  \rho^2}

    \rho^2=\dfrac12 r^2~\implies~ \rho=\dfrac r2 \cdot \sqrt{2}

    In your case \rho \approx 7.07



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  6. #6
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    Quote Originally Posted by KevinVM20 View Post
    It is all starting to click. However, I do not see how you got the 7.07.
    I have a pizza. The radius is 10 inches long. ...
    If \rho=\dfrac r2 \cdot \sqrt{2} and r = 10 then \rho=\dfrac{10}2 \cdot \sqrt{2} \approx 7.071067812...

    I rounded this value to \rho = 7.07
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  7. #7
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    Quote Originally Posted by KevinVM20 View Post
    4. What would the area of the whole pizza be if it were made of half pieces?
    Seems to me like a bit of a trick question; the total area is the total area regardless of how it is sliced. So it's still 100\pi, no?
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