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Thread: arithmetic series

  1. #1
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    arithmetic series

    q:
    Question: A set of steps for the end of a pier are built of stone. A sketch of the cross-section of the steps is shown.

    Each step has rise of 0.2m and tread of 0.6m. Form a series to calculate the area of the cross-section.

    arithmetic series-q1.jpg

    I tried to answer the question but I got it wrong
    I tried to to find the area of A
    using Sums of arithmetic - i found the length of the stairs
      S_{12}= \frac{12}{2}\left [ 2(0.6)+(12-1)0.2 \right ] = 19.8
    area of a =  \frac{1}{2}(19.8)(12) = 118.8
    area b = 8 x 12 = 96




    therefore total area = 214.8 square metres


    the answer in the book is 118.8 square metres

    Can someone please help
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  2. #2
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    Re: arithmetic series

    12/0.6 = 20 rectangular strips of width 0.6 m and variable height 8+0.2n meters

    $S_{20}=(8 \cdot 0.6) + [(8+0.2) \cdot 0.6] + [(8+0.4) \cdot 0.6] + ... + [(8+3.8) \cdot 0.6]$

    $S_{20} = 0.6(8+8.2+8.4+ ... +11.8) = 0.6\bigg[\dfrac{20}{2}(16+19 \cdot 0.2) \bigg] = 118.8 \, m^2$
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  3. #3
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    Re: arithmetic series

    In case this helps:

    h = height = 8
    w = width = 12
    So area without steps = hw = 96

    r = rise = .2
    t = tread = .6

    Arithmetic series:
    S = n[2a + d(n - 1)]/2
    a = 1st term = rt
    d = difference = rt (so a=d)
    n = number of terms = w/t - 1

    Area = hw + n[2a + d(n - 1)]/2 = 118.8 (as per Skeeter)

    NOTE:
    don't ask me to explain further: no blackboard
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