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    The strength of earthquakes is measured on the Richter magnitude scale, with valuestypically between 0 and 8 where 8 is the most severe. The Gutenberg–Richter equation gives the average number of earthquakes per year, N,which have a magnitude of at least M. For a particular region the equation is log10 N=a-M, for somea∈ℝ. This region has an average of 100 earthquakes per year with a magnitude of at least 3.The equation for this region can also be written asN=b10M.The expected length of time, in years, between earthquakes with a magnitude of atleast M is1N. Within this region the most severe earthquake recorded had a magnitude of 7.2.

    Question
    SLPaper 1

    The strength of earthquakes is measured on the Richter magnitude scale, with valuestypically between 0 and 8 where 8 is the most severe.

    The Gutenberg–Richter equation gives the average number of earthquakes per year, N,which have a magnitude of at least M. For a particular region the equation is

    log10 N=a-M, for somea∈ℝ.

    This region has an average of 100 earthquakes per year with a magnitude of at least 3.

    The equation for this region can also be written asN=b10M.

    The expected length of time, in years, between earthquakes with a magnitude of atleast M is1N.

    Within this region the most severe earthquake recorded had a magnitude of 7.2.

    1.

    Find the value of a.

    [2]
    Verified
    Solution

    log10 100=a-3 (M1)

    a=5 A1

    [2 marks]

    2.

    Find the value of b.

    [2]
    Verified
    Solution

    EITHER

    N=105-M (M1)

    =10510M=10000010M


    OR

    100=b103 (M1)


    THEN

    b=100000=105 A1

    [2 marks]

    3.

    Given 0<M<8, find the range for N.

    [2]
    Verified
    Solution

    0.001<N<10000010-3<N<105 A1A1


    Note: Award A1 for correct endpoints and A1for correct inequalities/interval notation.

    [2 marks]

    4.

    Find the expected length of time between this earthquake and the next earthquake ofat least this magnitude. Give your answer to the nearest year.

    [2]
    Verified
    Solution

    N=105107.2=0.0063095… (M1)

    length of time=10.0063095…=102.2

    =158years A1

    [2 marks]

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