# Series :) enjoy

• Nov 12th 2012, 06:37 AM
Redwood87
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. • Nov 12th 2012, 06:43 AM Prove It Re: Series :) enjoy Quote: Originally Posted by Redwood87 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.
• Nov 12th 2012, 06:52 AM
cac2008
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.
• Nov 12th 2012, 06:53 AM
Redwood87
Re: Series :) enjoy
how do you evaluate how many terms though?? thats the bit im stuck on..
• Nov 12th 2012, 08:13 AM
cac2008
Re: Series :) enjoy
1) Total volume = 16x9x9 = 1296 cubic metres = 1296000 litres

$\displaystyle a_{1}=150$

$\displaystyle a_{2}=350$

$\displaystyle a_{3}=350$

Common Difference = $\displaystyle a_{3}-a_{2}=a_{2}-a_{1}=200$

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

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

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

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

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

$\displaystyle 2592000-200n^{2}=100n$

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

$\displaystyle n \approx 113.59,-114.09$

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 $\displaystyle r=\frac{a_{n+1}}{a_{n}}$ and repeat, and in 3 rearrange the equation for $\displaystyle S_{n}$ to obtain n, and solve.