Answer:
4 hours
Step-by-step explanation:
Let x minutes be the time needed for the powerful pomp to fill the pool. Then x+120 minutes is the time needed to the other pomp to fill the pool. The powerful pomp fills [tex]\dfrac{1}{x}[/tex] of the pool in a minute and the other pomp fills [tex]\dfrac{1}{x+120}[/tex] of the pool in a minute. Filling together, they fill [tex]\dfrac{1}{x}+\dfrac{1}{x+120}[/tex] of the pool in a minute and
[tex]144\cdot \left(\dfrac{1}{x}+\dfrac{1}{x+120}\right)[/tex]
in 144 minutes.
Thus,
[tex]144\cdot \left(\dfrac{1}{x}+\dfrac{1}{x+120}\right)=1.[/tex]
Solve this equation:
[tex]\dfrac{144(x+120)+144x}{x(x+120)}=1,\\ \\144x+144\cdot 120+144x=x^2+120x,\\ \\x^2-168x-17280=0,\\ \\D=(-168)^2-4\cdot (-17280)=97344,\\ \\x_{1,2}=\dfrac{-(-168)\pm \sqrt{97344}}{2}=\dfrac{168\pm 312}{2}=-72,\ 240.[/tex]
Hence, it is needed the powerful pomp 240 minutes = 4 hours to fill the pool.
