Question

Two pipes are used to fill the astern completely. The second pipe opened one hour after the
first pipe. Three hours after the first plpe has opened, there is still (9/20) of the cistern is to
be filled. When the cistern is completely filled, it was found that each pipe has filled half of
the astern. How many hours would it take each one to completely filled the cistem
individually?

Answer 1

Answer:

       The first pipe would take 10 hours.

       The second pipe would take 8 hours.

Step-by-step explanation:

Let a be the fraction of the cistern filled by the first pipe in one hour.

Let b be the fraction of the cistern filled by the second pipe in one hour.

Let t be the time (in hours) taken to completely fill the cistern, from when the first pipe was opened.

Then...

    Three hours after the first pipe has opened, there is still (9/20) of the cistern to be filled

⇒  Three hours after the first pipe opened, the fraction 3a from the first pipe, plus the fraction 2b (only 2 hours) from the second pipe, together made 11/20 of the cistern

⇒  3a + 2b  =  11/20

⇒  60a + 40b = 11      ...(*)

Also...

    When the cistern is completely filled, it was found that each pipe has filled half of the cistern

⇒  t hours after the first pipe opened, the fraction at from the first pipe was ½, and the fraction b(t - 1) (only t-1 hours) from the second pipe was also ½

⇒  at = ½    and    b(t - 1) = ½

⇒  at = ½    and    bt = b + ½      ...(**)

Multiplying (*) by t and then using (**) to eliminate a gives

    60at + 40bt = 11t

⇒  60×½ + 40(b + ½) = 11t

⇒  30 + 40b + 20 = 11t

⇒  40b + 50 = 11t

Multiplying this by b and using (**) to eliminate t gives

    40b² + 50b = 11bt = 11(b + ½) = 11b + 11/2

⇒  40b² + 39b - 11/2 = 0.

Let's use the quadratic formula to solve this for b.

The discriminant is

   Δ  =  39² - 4×40×(-11/2)  =  39² + 40×22  =  2401  =  49²

So...

  b  =  ( -39 ± √(49²) ) / (2 × 40)  =  ( -39 ± 49 ) / 80

Since b must be positive, it follows that

   b  =  ( -39 + 49 ) / 80  =  10 / 80  ⇒  b = 1/8

Putting this into (*) to get a

   a  =  (11 - 40b) / 60  =  (11 - 5) / 60  = 6 / 60  ⇒  a = 1/10

So the first pipe would take 10 hours to fill the cistern alone,

and the second pipe would take 8 hours to fill the cistern alone.

Hope this helps.

Answer 2

Answer:

19

Step-by-step explanation: