Differential Equations
ODE application — mixture problem
Grade Class 12
Question:
<p>Brine tank: 100 L, initial concentration 2 kg/L. Pure water in at 5 L/min, out at 5 L/min. Find \\(Q(t)\\).</p>
<span>\(200e^{-t/20}\)</span>
<span>\(200e^{-t/5}\)</span>
<span>\(200e^{-5t}\)</span>
<span>\(100e^{-t/20}\)</span>
Step-by-Step Solution
Key Concept: dQ/dt = rate in - rate out = 0 - 5Q/100.
<div class='solution'><p>$\dfrac{dQ}{dt}=-\dfrac{5Q}{100}=-\dfrac{Q}{20}$. Separable: $\dfrac{dQ}{Q}=-\dfrac{dt}{20}$ → $Q=Q_0e^{-t/20}$. Initial: $Q(0)=200$ kg (100 L × 2 kg/L). $Q(t)=200e^{-t/20}$. <strong>Answer: (1)</strong>.</p><p class='key-concept'>🔑 Key Concept: Mixture problems: dQ/dt = (rate in × conc in) - (rate out × conc out). With constant volume: conc out = Q/V.</p></div>
Correct Answer: 1