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Due to the large potato harvest this year, the farm bought a new, more powerful sorter in addition to the old one. Now both machines work simultaneously, and therefore the daily harvest is processed in 12 hours. If only the old machine were working, it would need 10 hours more than the new machine to process the daily harvest. How long would it take the old sorter?

Correct answer:

a =  30 h

Step-by-step explanation:

$$\begin{gathered} T=12\text{h} \\ a=10+b \\ T(1/a+1/b)=1 \\ T(1/a+1/(a-10))=1 \\ T(a-10)+T\cdot a=(a-10)\cdot a \\ 12(a-10)+12\cdot a=(a-10)\cdot a \\ -a^2+34a-120=0 \\ {a}^2-34a+120=0 \\ p=1;q=-34;r=120 \\ D=q^2-4pr=34^2-4\cdot 1\cdot 120=676 \\ D>0 \\ {a}_{1,2}=\frac{-q\pm \sqrt{D}}{2p}=\frac{34\pm \sqrt{676}}{2} \\ {a}_{1,2}=\frac{34\pm 26}{2} \\ {a}_{1,2}=17\pm 13 \\ {a}_1=30 \\ {a}_2=4 \\ a>10;a>10h \\ a=a_1=30=30\text{h} \\ b=a-10=30-10=20\text{h} \end{gathered}$$

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