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Thread: Radius Increase

  1. #1
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    Radius Increase

    The radius of a circle is r units. By how many units should the radius be increased so that the area increases by b square units?

    I think the area of a circle formula is needed.

    A = (pi)r^2

    How do I get started?
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  2. #2
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    Re: Radius Increase

    Quote Originally Posted by mathdad1965 View Post
    The radius of a circle is r units. By how many units should the radius be increased so that the area increases by b square units?

    I think the area of a circle formula is needed.

    A = (pi)r^2

    How do I get started?
    You're getting kind of comfortable in asking for help before you start the problem. Let's try this: You have the area formula and you want to increase it by b units. So the new area is A + b, right? Can you go from there?

    In the future, please let us know what work you've been able to do, or at least give some indication of what you have tried.

    -Dan
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  3. #3
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    Re: Radius Increase

    Quote Originally Posted by topsquark View Post
    You're getting kind of comfortable in asking for help before you start the problem. Let's try this: You have the area formula and you want to increase it by b units. So the new area is A + b, right? Can you go from there?

    In the future, please let us know what work you've been able to do, or at least give some indication of what you have tried.

    -Dan
    Do I also increase the radius by b units?

    (A + b) = pi(r + b)^2

    Do I now solve for r?
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  4. #4
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    Re: Radius Increase

    No, the area is to increase by b square units. That does NOT mean the radius will increase by b units. If we take the original radius t be r then we have A= \pi r^2. If the new radius, such that the area is increased by b square units, is r' then we have A+ b= \pi r'^2. You want to find r'- r.
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  5. #5
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    Re: Radius Increase

    Quote Originally Posted by HallsofIvy View Post
    No, the area is to increase by b square units. That does NOT mean the radius will increase by b units. If we take the original radius t be r then we have A= \pi r^2. If the new radius, such that the area is increased by b square units, is r' then we have A+ b= \pi r'^2. You want to find r'- r.
    Are you saying to take the derivative of r?
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  6. #6
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    Re: Radius Increase

    Quote Originally Posted by mathdad1965 View Post
    Are you saying to take the derivative of r?
    NO!.
    You know, you should have made up a problem yourself:
    as example, take a circle radius 5 and a circle radius 12;
    then the radius difference is 7.
    Calculate the area of both circles, then work backward
    with the area difference and the radius increase:
    means you'll be able to "see" what's going on, instead of
    making wild guesses backed by no work!

    Using the above example:
    p = pi
    r = initial radius (5)
    d = area difference (~373.85)
    x = radius increase (?)

    p(r + x)^2 - p(r^2) = d
    Solve resulting quadratic for x : you'll get x = 7.
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  7. #7
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    Re: Radius Increase

    Quote Originally Posted by DenisB View Post
    NO!.
    You know, you should have made up a problem yourself:
    as example, take a circle radius 5 and a circle radius 12;
    then the radius difference is 7.
    Calculate the area of both circles, then work backward
    with the area difference and the radius increase:
    means you'll be able to "see" what's going on, instead of
    making wild guesses backed by no work!

    Using the above example:
    p = pi
    r = initial radius (5)
    d = area difference (~373.85)
    x = radius increase (?)

    p(r + x)^2 - p(r^2) = d
    Solve resulting quadratic for x : you'll get x = 7.
    What's with the NO!? Please, skip my questions. Are you that upset with my questions?
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  8. #8
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    Re: Radius Increase

    Quote Originally Posted by mathdad1965 View Post
    What's with the NO!? Please, skip my questions. Are you that upset with my questions?
    So you're allowed to spout off at us when you don't like our answers but no one is allowed to spout off at you when they feel you aren't paying attention.

    I see.
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  9. #9
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    Re: Radius Increase

    Quote Originally Posted by romsek View Post
    So you're allowed to spout off at us when you don't like our answers but no one is allowed to spout off at you when they feel you aren't paying attention.

    I see.
    I am doing the best I can to understand everyone who helps me here.
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  10. #10
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    Re: Radius Increase

    The radius of a circle is r units. By how many units should the radius be increased so that the area increases by b square units?

    Let $\Delta r =$ radius increase

    $A = \pi r^2$

    $A+b = \pi(r+\Delta r)^2$

    $\dfrac{A+b}{\pi} = (r+\Delta r)^2$

    $\dfrac{A+b}{\pi} = r^2 + 2r \cdot \Delta r + (\Delta r)^2$

    $0 = (\Delta r)^2 + 2r \cdot \Delta r + r^2 - \dfrac{A+b}{\pi}$

    $0 = (\Delta r)^2 + 2r \cdot \Delta r + r^2 - \dfrac{A+b}{\pi}$


    $\Delta r = \dfrac{-2r + \sqrt{(2r)^2 - 4\left(r^2 - \dfrac{A+b}{\pi}\right)}}{2}$

    $\Delta r = \dfrac{-2r + \sqrt{4r^2 - 4r^2 + \dfrac{4(A+b)}{\pi}}}{2}$

    $\Delta r = \sqrt{\dfrac{(A+b)}{\pi}}-r$
    Last edited by skeeter; Feb 6th 2017 at 06:43 PM.
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  11. #11
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    Re: Radius Increase

    $A=\pi r^2$

    $A+b=\pi R^2 \implies R= \sqrt{\dfrac{A+b}{\pi}}$

    $\Delta r = R-r = \sqrt{\dfrac{A+b}{\pi}} - r$

    edit ... took the LONG road to Dallas the first time.
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  12. #12
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    Re: Radius Increase

    Quote Originally Posted by skeeter View Post
    $A=\pi r^2$

    $A+b=\pi R^2 \implies R= \sqrt{\dfrac{A+b}{\pi}}$

    $\Delta r = R-r = \sqrt{\dfrac{A+b}{\pi}} - r$

    edit ... took the LONG road to Dallas the first time.
    Very clear calculation.
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  13. #13
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    Re: Radius Increase

    Very nice 'n compact, Skeeter.
    Takes about half the time as mine
    Thanks from mathdad1965
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