Two brine tanks connected as shown in the fig. Water flows into the top of tank $T_{1}$ at rate of $15 l / \min$ . The brine solution is pumped from tank $T_{1} \to T_{2}$ at the rate of $40 l / \min$ and from tank $T_{2} \to T_{1}$ at the rate of $25 l / \min$ . The brine solution also flows out of tank $T_{2}$ at the rate of $15 l / \min$ .
Suppose there is $100 l$ of brine in tank $T_{1}$ and $120 l$ of brine in tank $T_{2}$ at the beginning. Let ‘x’ kg be amount of salt in tank $T_{1}$ after ‘t’ minutes, and ‘y’ be that in tank $T_{2}$ . If each tank is mixed perfectly throughout the whole time, then which of the following options is/are correct?

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$\frac{d x}{d t} = \frac{- y}{3} + \frac{2 x}{5}$

$\frac{d y}{d t} = \frac{- y}{3} + \frac{2 x}{5}$

$\frac{d x}{d t} = \frac{5 y}{24} - \frac{2 x}{5}$

$\frac{d y}{d t} = \frac{5 y}{24} + \frac{2 x}{5}$

answer is A, D.

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Detailed Solution

$\frac{d x}{d t} = T_{1} \to T_{2} \frac{- 40 x}{100} + T_{2} \to T_{1} \frac{25 y}{120} = \frac{5 y}{24} - \frac{2 x}{5} \frac{d y}{d t} = \frac{40 x}{100} T_{1} \to T_{2} - \frac{25 + 15}{120} y T_{2} \to T_{1} T_{2} \to o u t = \frac{2 x}{5} - \frac{y}{3}$