Let's go step-by-step for each part of the problem: --- ### **Given:** - Tank volume: **200 L** - Initial brine: **100 L** with **10 kg** of dissolved salt - Brine enters at **10 L/min** with **0.1 kg/L** salt - Brine leaves at **5 L/min** - Tank is perfectly mixed Let \( y(t) \) = amount of salt (kg) in the tank at time \( t \) (minutes). --- ## **(a) Differential Equation for \( y(t) \)** ### **Step 1: Salt Inflow Rate** - Inflow rate of brine: \( 10 \) L/min - Concentration of inflowing brine: \( 0.1 \) kg/L \[ \text{Salt in per minute} = (10 \text{ L/min}) \times (0.1 \text{ kg/L}) = 1 \text{ kg/min} \] --- ### **Step 2: Salt Outflow Rate** - Outflow rate of brine: \( 5 \) L/min - Tank is well-mixed, so concentration in tank at time \( t \) is \( \frac{y(t)}{V(t)} \) where \( V(t) \) is the volume at time \( t \). Volume at time \( t \): \[ V(t) = \text{Initial volume} + (\text{in rate} - \text{out rate}) \times t = 100 + (10 - 5)t = 100 + 5t \] So, salt out per minute: \[ \text{Salt out per minute} = (\text{outflow rate}) \times \text{concentration in tank} = 5 \times \frac{y(t)}{100 + 5t} \] --- ### **Step 3: Differential Equation** Change in salt in the tank: \[ \frac{dy}{dt} = \text{salt in} - \text{salt out} = 1 - 5 \frac{y(t)}{100 + 5t} \] **Initial condition**: at \( t = 0 \), \( y(0) = 10 \) kg --- ### **Final Answer (a):** \[ \boxed{ \frac{dy}{dt} = 1 - \frac{5y}{100 + 5t}, \quad y(0) = 10 } \] --- ## **(b) Type of Differential Equation** - It can be rewritten as: \[ \frac{dy}{dt} + \frac{5}{100+5t}y = 1 \] - This is a **first-order linear** ODE (of the form \( y' + p(t)y = q(t) \)) - It is also **separable** (can solve for \( y \) and \( t \) on either side) --- ### **Final Answer (b):** \[ \boxed{\text{Both separable and first-order linear}} \] --- ## **(c) Time for Tank to Overflow** Overflow happens when the tank reaches **200 L** Recall: \[ V(t) = 100 + 5t \] Set \( V(t) = 200 \): \[ 100 + 5t = 200 \implies 5t = 100 \implies t = 20 \] --- ### **Final Answer (c):** \[ \boxed{20\ \text{minutes}} \] --- ### **Summary of Answers:** **(a)** \( \frac{dy}{dt} = 1 - \frac{5y}{100 + 5t},\quad y(0) = 10 \) **(b)** Both separable and first-order linear **(c)** 20 minutes to overflow