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Math Help - Geometric progression

  1. #1
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    Geometric progression

    A roll of thread with 90pi cm long is cut into 5 parts to make 5 circles.The radii of the circles increase by 1 cm consecutively.Calculate

    a)the radius of the smallest circle
    b)the circumference of the 6 th circle
    c)the number of circles obtained if the original length of the thread is 660pi cm
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  2. #2
    Master Of Puppets
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    Hi there mastermin346

    Quote Originally Posted by mastermin346 View Post
    A roll of thread with 90pi cm long is cut into 5 parts to make 5 circles.The radii of the circles increase by 1 cm consecutively.Calculate

    a)the radius of the smallest circle
    Well i'm reading this as the length of string is 90cm?

    Therefore

    90 = length of string #1 + length of string #2+ .... +length of string #5

    And the difference in each length, shaped as a circle is, adding 1 to the next radius so

    90 = 2\pi r+2\pi (r+1)+2\pi (r+2)+2\pi (r+3)+2\pi (r+4)

    Factoring out and then dividing 2\pi

    90 = 2\pi (r+ (r+1)+ (r+2)+ (r+3)+ (r+4))

    \frac{90}{2\pi} =  (r+ (r+1)+ (r+2)+ (r+3)+ (r+4))

    Grouping like terms on the RHS

    \frac{90}{2\pi} =  5r+10

    \frac{45}{\pi} =  5r+10

    \frac{45}{\pi} -10=  5r

    r= \frac{\frac{45}{\pi} -10}{5}

    Quote Originally Posted by mastermin346 View Post
    b)the circumference of the 6 th circle
    c)the number of circles obtained if the original length of the thread is 660pi cm
    You should be able to use the same logic to finish the rest.
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  3. #3
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    Hello mastermin346
    Quote Originally Posted by mastermin346 View Post
    A roll of thread with 90pi cm long is cut into 5 parts to make 5 circles.The radii of the circles increase by 1 cm consecutively.Calculate

    a)the radius of the smallest circle
    b)the circumference of the 6 th circle
    c)the number of circles obtained if the original length of the thread is 660pi cm
    A number of points:

    1 There's no reason to suppose there was a typo in the original question: the length of the thread could quite well be 90\pi cm; in which case, the radius of the smallest circle is 7 cm.

    2 This is an example of an arithmetic progression, not a geometric progression.

    3 For part (c), if the initial radius is 7 cm and there are n circles altogether, then the total length of thread used is:
    2\pi\big(7+(7+1) + (7+2) + ... + (7+[n-1])\big)
    =2\pi\times \tfrac12n(13+n), using the AP formula S = \tfrac12n(a+l)

    =660\pi
    This gives a quadratic equation for n with one positive root.
    Spoiler:
    n=20

    Grandad
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