Salt Leaving Rate 863010

1. **State the problem:** We want to find the rate at which salt leaves the tank at time $t$. 2. **Recall the volume of brine in the tank at time $t$:** From previous part, volume is $V(t) = 100 + t$ gallons. 3. **Concentration of salt in the tank at time $t$:** Let $y(t)$ be the amount of salt (in pounds) at time $t$. The concentration is salt amount divided by volume: $$\text{Concentration} = \frac{y(t)}{100 + t} \text{ lb/gal}$$ 4. **Outflow rate of brine:** The brine flows out at 5 gal/min. 5. **Rate salt leaves the tank:** Rate out = (concentration) $\times$ (outflow rate) $$\text{Rate out} = \frac{y(t)}{100 + t} \times 5 = \frac{5y(t)}{100 + t} \text{ lb/min}$$ **Final answer:** The salt leaves the tank at rate $$\frac{5y(t)}{100 + t}$$ pounds per minute at time $t$.