If two pipes function simultaneously , the reservoir will be filled in 12 hours. One pipe fills the reservoir 10 hours faster than the other. How many hoirs does the second pipe take to fill the reservoir
Answers
Suppose there are two pipes names A and B
It is given that pipe B is faster than pipe A
So time taken by pipe B is 10 hour less than time taken by pipe A
Suppose pipe A alone can fill it in x hours and pipe B alone can fill it in x-10 hours
The one hour work of Pipe A = 1/x and 1 hour work of pipe B = 1/x-10
When both pipe work together the one hour work = (1/x) + 1/(x-10 )
it is given that both togther take 12 hour that means together one hor work should be = 1/12
This suggest : ( 1/x) + (1/(x-10) = 1/12
When we take lcd and simplify it becomes :
(x-10 + x) 1
-------------- = ---
x( x-10) 12
2x - 10 1
-------- = -------
x(x-10 ) 12
When we cross multiply it becomes :
-120= x(x-10 )
24x - 120 = x^2 - 10x
x^2 - 34x +120 = 0
Let us tr use quadratic formula a= 1 b= -34 c= 120
we plug them in the formula : -b +/- ( b^2 - 4ac)^(1/2)
--------------------------------------
2a
On plugging these number in the formula we get : x =30
Answer :
So the time taken by pipe B(faster pipe) = 30-10 =20hours
SOLUTION :
Let the faster pipe fill the reservoir in x h.
Then, the slower pipe the reservoir in (x + 10) h
In 1 hour the faster pipe fills the portion of the reservoir : 1/x
In 12 hour the faster pipe fills the portion of the reservoir : 12 × 1/x = 12/x
In 1 hour the slower pipe fills the portion of the reservoir : 1/(x + 10)
In 12 hour the faster pipe fills the portion of the reservoir : 12 × 1/(x + 10) = 12/(x +10)
A.T.Q
12/x + 12/(x +10) = 1
12(1/x + 1/(x + 10) ) = 1
1/x + 1/(x + 10) = 1/12
(x + 10 + x ) / [x(x + 10)] = 1/12
[By taking LCM]
2x +10 /(x² + 10x) = 1/12
x² + 10x = 12(2x +10)
x² + 10x = 24x + 120
x² + 10x - 24x - 120 = 0
x² - 14x - 120 = 0
x² + 6x - 20x - 120 = 0
[By splitting middle term]
x(x + 6) - 20(x + 6) = 0
(x - 20) (x + 6) = 0
(x - 20) or (x + 6) = 0
x = 20 or x = - 6
Since, time cannot be negative, so x ≠ - 6
Therefore, x = 20
The faster pipe takes 20 hours to fill the reservoir
Hence, the slower pipe (second) takes (x + 20) = 30 hours to fill the reservoir.
HOPE THIS ANSWER WILL HELP YOU…...