Pumps

A tank is filled by two pumps working together in 23 minutes. The first pump alone fills the tank 37 minutes faster than the second pump alone. How long does it take the first pump to fill the tank on its own?

Final Answer:

t₁ =  34 min

Step-by-step explanation:

$$\begin{gathered} t=23\text{min} \\ {t}_1=t_2-37\text{min} \\ t\cdot (Q_1+Q_2)=1 \\ {Q}_1\cdot t_1=1 \\ {Q}_2\cdot t_2=1 \\ t\cdot (1/t_1+1/t_2)=1 \\ {t}_1=t_2-37 \\ t\cdot (t_2+t_1)=t_1\cdot t_2 \\ t\cdot (T+(T-37))=(T-37)\cdot T \\ 23\cdot (T+(T-37))=(T-37)\cdot T \\ -T^2+83T-851=0 \\ {T}^2-83T+851=0 \\ a=1;b=-83;c=851 \\ D=b^2-4ac=83^2-4\cdot 1\cdot 851=3485 \\ D>0 \\ {T}_{1,2}=\frac{-b\pm \sqrt{D}}{2a}=\frac{83\pm \sqrt{3485}}{2} \\ {T}_{1,2}=41.5\pm 29.516944 \\ {T}_1=71.016944286 \\ {T}_2=11.983055714 \\ {t}_2=T_1=71.0169\doteq 71.0169\text{min} \\ {t}_1=t_2-37=71.0169-37=34\text{min} \end{gathered}$$

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