Math

The table below shows the number of molecules in experiment B.

Time(h) : 1st , 2nd , 3rd , 4th , 5th
No. of molecules : 3 , 8 , 15 , 24, 35

(i) how many molecules will there be in nth hour?
(ii) from which hour onwards would the no. of molecules be in experiment B?

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  1. I tried an exponential and that did not work so try a polynomial:

    change in n per hour 5 7 9 11
    change in change in n per hour = 2
    constant second derivative --> parabola

    n = a + k t + c t^2
    dn/dt = k + 2 c t
    d^2n/dt^2 = 2 c
    so
    2c = 2
    c = 1
    and
    n = a + k t + t^2

    when t = 1, n = 3
    3 = a + k + 1
    a + k = 2
    when t = 2, n = 8
    8 = a + 2 k + 4
    a + 2 k = 4
    subtract those 2 equations
    -k = - 2
    k = 2
    so
    n = a + 2 t + t^2

    when t = 3, n = 15
    15 = a + 2(3) + 9
    a = 0
    so
    n = + 2 t + t^2

    check when t = 5
    n = 10 + 25 = 35 check

    so I claim that
    n = 2 t + t^2

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