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Math Help - Series :) enjoy

  1. #1
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    Series :) enjoy

    1) Water fills a tank at a rate of 150 litres during the first hour, 350 litres during the second, 550 during the third and so on. Find the number of hours necessary to fill a rectangular tank 16m x 9m x 9m.


    2) A firm starts work with 110 employees for the 1st week. the number of the employees rises by 6% for the first week. how many persons will be employed in the 20th week if the present rate of expansion continues.

    3) a contractor hires out machinery. in the first year of hiring out one piece of equipment the profit is $6000, but this diminishes by 5% in successive years. show that the annual profits form a geometric progression and find the total of all the profits for the first 5 years.
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  2. #2
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    Re: Series :) enjoy

    Quote Originally Posted by Redwood87 View Post
    1) Water fills a tank at a rate of 150 litres during the first hour, 350 litres during the second, 550 during the third and so on. Find the number of hours necessary to fill a rectangular tank 16m x 9m x 9m.


    2) A firm starts work with 110 employees for the 1st week. the number of the employees rises by 6% for the first week. how many persons will be employed in the 20th week if the present rate of expansion continues.

    3) a contractor hires out machinery. in the first year of hiring out one piece of equipment the profit is $6000, but this diminishes by 5% in successive years. show that the annual profits form a geometric progression and find the total of all the profits for the first 5 years.
    The first is an arithmetic sequence. Evaluate however many terms are needed in your arithmetic series to get a capacity (sum) of 1 296 000 L.
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    Re: Series :) enjoy

    Second is a geometric series: determine Sum to however many people are present
    Third is also a geometric series. Demonstrate equal ratios between successive terms and compute the sum.
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    Re: Series :) enjoy

    how do you evaluate how many terms though?? thats the bit im stuck on..
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    Re: Series :) enjoy

    1) Total volume = 16x9x9 = 1296 cubic metres = 1296000 litres

    a_{1}=150

    a_{2}=350

    a_{3}=350

    Common Difference = a_{3}-a_{2}=a_{2}-a_{1}=200

    a_{n}=a_{1}+(n-1)d ==> a_{n}=150+200(n-1)

    Sum to n-terms: S_{n}=\frac{n}{2}(2a+(n-1)d)

    \uptherefore 2592000=n(300+200n-200)

    \frac{2592000}{n}-200n=100

    \frac{2592000-200n^{2} }{n}=100

    2592000-200n^{2}=100n

    -200n^{2}-100n+2592000=0

    n \approx 113.59,-114.09

    As n must be positive, n=113.59 or 114 (if rounding up)

    For 2, use the same process as above, simply replacing the common difference, d, with the common ratio, r such that r=\frac{a_{n+1}}{a_{n}} and repeat, and in 3 rearrange the equation for S_{n} to obtain n, and solve.
    Last edited by cac2008; November 12th 2012 at 08:18 AM. Reason: Naff Latex skills!
    Thanks from Redwood87
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