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    At a given instant, the radii of two concentric circles are 8 cm and 12 cm. The radius of the outer circle is increasing at a rate of 1cm/s and the radius of the inner circle is increasing at a rate of 2cm/s. Find the rate of change of the area contained within the circles.
    ______________________

    Not 100% sure where to start.
    At first I tried to find the rate of change of the areas of the circles individually and take one away from the other.
    But I realise now it's going to be more complicated than that as at a certain point the inner over takes the outer and then the area trapped between the circles will begin to increase.

    Would appreciate it if someone could point me in the right direction
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    So we have area = π*r^2
    So the area enclosed would be equal to π(a^2-b^2) <--- calling a the radius of the outer circle, b the inner.
    dA/dt = π(2a*(da/dt) - 2b*(db/dt))

    Find the values for da/dt and db/dt when a = 12 and b = 8
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    da/dt(12) = 2*π*12 = 24π
    db/dt(8) = 2*π*8 = 16π

    Now, the outer radius (a) increases at a rate of 1cm/s and inner radius (b) increases at a rate of 2cm/s.

    dA/dt = π((24*1cm/s) - (16*2cm/s))
    dA/dt = π(24-32)
    = -8π cm^2 /s

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    (Original post by FryOfTheMann)
    So we have area = π*r^2
    So the area enclosed would be equal to π(a^2-b^2) <--- calling a the radius of the outer circle, b the inner.
    dA/dt = π(2a*(da/dt) - 2b*(db/dt))

    Find the values for da/dt and db/dt when a = 12 and b = 8
    Spoiler:
    Show






    da/dt(12) = 2*π*12 = 24π
    db/dt(8) = 2*π*8 = 16π

    Now, the outer radius (a) increases at a rate of 1cm/s and inner radius (b) increases at a rate of 2cm/s.

    dA/dt = π((24*1cm/s) - (16*2cm/s))
    dA/dt = π(24-32)
    = -8π cm^2 /s





    I'm following you to the 4th line. Why not just multiply the two terms by da/dt and db/dt if we know what they are?? We're told in the question that their rates of change are 1 and 2??
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    At a given instant, the radii of two concentric circles are 8 cm and 12 cm. The radius of the outer circle is increasing at a rate of 1cm/s and the radius of the inner circle is increasing at a rate of 2cm/s. Find the rate of change of the area contained within the circles.
    ______________________

    Not 100% sure where to start.
    At first I tried to find the rate of change of the areas of the circles individually and take one away from the other.
    But I realise now it's going to be more complicated than that as at a certain point the inner over takes the outer and then the area trapped between the circles will begin to increase.

    Would appreciate it if someone could point me in the right direction
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    (Original post by Retsek)
    I'm following you to the 4th line. Why not just multiply the two terms by da/dt and db/dt if we know what they are?? We're told in the question that their rates of change are 1 and 2??
    We know that at some point in time t=T we have two circles of radii a=8 and b=12. We also know that at this time \dfrac{da}{dt}=2 and \dfrac{db}{dt}=1.

    The area between the two is given by A=\pi(b^2-a^2) which implies that \dfrac{dA}{dt}=\pi(2b \dfrac{db}{dt} - 2a \dfrac{da}{dt}). So at the instant t=T we simply need to substitute in the info we are given.
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    (Original post by RDKGames)
    We know that at some point in time t=T we have two circles of radii a=8 and b=12. We also know that at this time \dfrac{da}{dt}=2 and \dfrac{db}{dt}=1.

    The area between the two is given by A=\pi(b^2-a^2) which implies that \dfrac{dA}{dt}=\pi(2b \dfrac{db}{dt} - 2a \dfrac{da}{dt}). So at the instant t=T we simply need to substitute in the info we are given.
    Okay so the rates at which the radii are changing are only at that instant? I thought the 1 and 2 cm/s were constant?
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    (Original post by Retsek)
    Okay so the rates at which the radii are changing are only at that instant? I thought the 1 and 2 cm/s were constant?
    Yes, everything you're calculating is happening at a certain instant.
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    (Original post by RDKGames)
    Yes, everything you're calculating is happening at a certain instant.
    But the question says "At a given instant, the radii of two concentric circles are 8cm and 12cm." There's a full stop before it continues to say "The radius of the outer circle is increasing...." So the given instant is only applicable to the radii at the instant not the rate of change??

    Also don't get how you say that we can find the rate of change as a function? We're told that at a given instant it equals 1 and 2,so surely that could be represented by any two functions that have values at 1 and 2??

    Sorry if these are dumb questions.
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    (Original post by Retsek)
    But the question says "At a given instant, the radii of two concentric circles are 8cm and 12cm." There's a full stop before it continues to say "The radius of the outer circle is increasing...." So the given instant is only applicable to the radii at the instant not the rate of change??

    Also don't get how you say that we can find the rate of change as a function? We're told that at a given instant it equals 1 and 2,so surely that could be represented by any two functions that have values at 1 and 2??

    Sorry if these are dumb questions.
    You're looking too much into it, and your questions dont make much sense. Everything is happening at an instant here.
 
 
 
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