question_answer

To fill a swimming pool two pipes are to be used. If the pipe of larger diameter is used for 4 hours and the pipe of smaller diameter for 9 hours, only half 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.

Answer:

Let the pipe of larger diameter fills the pool in x hours.

Then, the pipe with smaller diameter fills the pool in \[(x+10)\] hours.

According to the condition,

\[\frac{4}{x}+\frac{9}{x+10}=\frac{1}{2}\]

\[\Rightarrow \frac{4(x+10)+9x}{x(x+10)}=\frac{1}{2}\]

\[\Rightarrow 2(4x+40+9x)={{x}^{2}}+10x\]

\[\Rightarrow 8x+80+18x={{x}^{2}}+10x\]

\[\Rightarrow 26x+80={{x}^{2}}+10x\]

\[\Rightarrow {{x}^{2}}-16x-80=0\]

\[\Rightarrow {{x}^{2}}-20x+4x-80=0\]

\[\Rightarrow x(x-20)+4(x-20)=0\]

\[\Rightarrow (x-20)(x+4)=0\]

\[\Rightarrow x=20\]                                           [As \[x\ne -4\]]

Hence, the pipe with larger diameter fills the tank in 20 hours.

And, the pipe with smaller diameter fills the tank in 30 hours.