2 pipes

2 pipes can fill a tank in 35 minutes. The larger pipe alone can fill the tank in 24 minutes less time than the smaller pipe. How long does each pipie take to fill the tank alone?

Result

a =  60 min
b =  84 min

Solution:

$35 \cdot \ (1/a + 1/b)=1 \ \\ a=b - 24 \ \\ 35 \cdot \ (1/a + 1/(a+24))=1 \ \\ \ \\ 35 \cdot \ (a+24 + a)=a \cdot \ (a+24) \ \\ -a^2 +46a +840=0 \ \\ a^2 -46a -840=0 \ \\ \ \\ p=1; q=-46; r=-840 \ \\ D=q^2 - 4pr=46^2 - 4\cdot 1 \cdot (-840)=5476 \ \\ D>0 \ \\ \ \\ a_{1,2}=\dfrac{ -q \pm \sqrt{ D } }{ 2p }=\dfrac{ 46 \pm \sqrt{ 5476 } }{ 2 } \ \\ a_{1,2}=\dfrac{ 46 \pm 74 }{ 2 } \ \\ a_{1,2}=23 \pm 37 \ \\ a_{1}=60 \ \\ a_{2}=-14 \ \\ \ \\ \text{ Factored form of the equation: } \ \\ (a -60) (a +14)=0 \ \\ \ \\ a=a_{1}=60 \ \text{min}$

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$b=a + 24=60 + 24=84 \ \text{min}$

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