To fill a swimming pool two pipes are used. If the pipe of larger diameter used for 4 hours and the pipe of smaller diameter for 9 hours, only half of the pool can be filled. Find, how long it would take for each pipe to fill the pool separately, if the pipe of smaller diameter takes 10 hours more than the pipe of larger diameter to fill the pool?
Let the larger diameter pipe fill it in ‘a’ hours
The smaller diameter pipe fills it in ‘a + 10’ hours
In 1 hour, larger diameter pipe fills 1/a part of the pool.
In 1 hour, smaller diameter pipe fills 1/(a + 10) part of the pool.
Given, the pipe of larger diameter used for 4 hours and the pipe of smaller diameter for 9 hours, only half of the pool can be filled.
⇒ 4 × 1/a + 9 × 1/(a+10) = 1/2
⇒ 2(4a + 40 + 9a) = a2 + 10a
⇒ 26a + 80 = a2 + 10a
⇒ a2 – 16a – 80 = 0
⇒ a2 – 20a + 4a – 80 = 0
⇒ a(a – 20) + 4(a – 20) = 0
⇒ (a + 4)(a – 20) = 0
⇒ a = 20 hours
Time in which smaller diameter pipe fills the pool = 20 + 10 = 30 hours