SOLUTION: Water fills a tank at a rate of 150 litres during the first hour, 350 during the second hour, 550 litres during the third hour and so on. Find the number of hours necessary to fill

Algebra ->  Quadratic Equations and Parabolas  -> Quadratic Equations Lessons  -> Quadratic Equation Lesson -> SOLUTION: Water fills a tank at a rate of 150 litres during the first hour, 350 during the second hour, 550 litres during the third hour and so on. Find the number of hours necessary to fill      Log On


   

Question 982125: Water fills a tank at a rate of 150 litres during the first hour, 350 during the second hour, 550 litres during the third hour and so on. Find the number of hours necessary to fill a rectangular tank 16m x 9m x 9m.
I answered the question with an Arithmetic Series, but I need it to be a Quadratic Equation, but unsure how to plug the numbers in.
Answer by macston(5194) About Me  (Show Source):
You can put this solution on YOUR website!
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Tank volume=(16m)(9m)(9m)(1000 liters/m^3)=1296000 liters
S=sum(tank volume)=; a=first term=150 liters; d=common difference=200 liters
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S=%28n%2F2%29%28a%2B%28a%2B%28n-1%29d%29%29
1296000=%28n%2F2%29%28150%2B%28150%2B%28n-1%29200%29%29
1296000=%28n%2F2%29%28150%2B%28150%2B200n-200%29%29
1296000=%28n%2F2%29%28200n%2B100%29
1296000=n%28100n%2B50%29
1296000=100n%5E2%2B50n
25920=2n%5E2%2Bn
0=2n%5E2%2Bn-25920
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 2x%5E2%2B1x%2B-25920+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%281%29%5E2-4%2A2%2A-25920=207361.

Discriminant d=207361 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-1%2B-sqrt%28+207361+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%281%29%2Bsqrt%28+207361+%29%29%2F2%5C2+=+113.592270269
x%5B2%5D+=+%28-%281%29-sqrt%28+207361+%29%29%2F2%5C2+=+-114.092270269

Quadratic expression 2x%5E2%2B1x%2B-25920 can be factored:
2x%5E2%2B1x%2B-25920+=+2%28x-113.592270269%29%2A%28x--114.092270269%29
Again, the answer is: 113.592270269, -114.092270269. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+2%2Ax%5E2%2B1%2Ax%2B-25920+%29

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So n=113.59 hours