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.
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