calculus help please?

word problem help please

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  • Lopez
    Lv 5
    4 years ago
    Best Answer

    dθ / dt = ( degree for 1 rotation ) / ( period for 1 rotation ) = 2π / 3

    By the pic. below, we have

    L = 16 * tan θ

    dL / dt

    = 16 * d tan θ / dt

    = 16 * sec^2 θ * ( dθ / dt )

    = 16 * sec^2 θ * ( 2π / 3 )

    = ( 32π / 3 ) * sec^2 θ

    When θ = 15°

    dL / dt

    = ( 32π / 3 ) * sec^2 15°

    = ( 32π / 3 ) * ( 1 + tan^2 15° )

    = ( 32π / 3 ) * ( 1 + 7 - 4√3 ) ..... see the note below

    = ( 32π / 3 ) * ( 8 - 4√3 ) ..... Ans

    Note.

    tan^2 15° = 7 - 4√3

    pf :

    By the formula :

    tan ( β / 2 ) = ± √ [ ( 1 - cos β ) / ( 1 + cos β ) ]

    So ,

    tan^2 ( β / 2 ) = ( 1 - cos β ) / ( 1 + cos β )

    Let β = 30° , we have

    tan^2 15°

    = ( 1 - cos 30° ) / ( 1 + cos 30° )

    = [ 1 - ( √3 / 2 ) ] / [ 1 + ( √3 / 2 ) ]

    = [ ( 2 - √3 ) / 2 ] / [ ( 2 + √3 ) / 2 ]

    = ( 2 - √3 ) / ( 2 + √3 )

    = [ ( 2 - √3 )( 2 - √3 ) ] / [ ( 2 + √3 )( 2 - √3 ) ]

    = ( 2 - √3 )^2

    = 7 - 4√3

    Attachment image
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