Differential Equations
Applications — draining tank model
Grade Class 12
Question:
<p>Water is drained from a vertical cylindrical tank. The rate of drop of water level: \(\dfrac{dy}{dt} = -k\sqrt{y}\). Given \(y(0) = 4\). Find time to drain (\(y=0\)).</p>
<span>\(30\text{ min}\)</span>
<span>\(45\text{ min}\)</span>
<span>\(60\text{ min}\)</span>
<span>\(t = 4/k\)</span>
Step-by-Step Solution
Key Concept: Separate variables: dy/\sqrt{y} = -k dt, then integrate.
<div class='solution'><p><strong>Step 1:</strong> \(\dfrac{dy}{dt} = -k\sqrt{y}\). Separate:</p>
<p>\[\frac{dy}{\sqrt{y}} = -k\,dt \implies 2\sqrt{y} = -kt + C\]</p>
<p><strong>Step 2:</strong> At \(t=0, y=4\): \(C = 2\sqrt{4} = 4\). So \(2\sqrt{y} = 4 - kt\).</p>
<p><strong>Step 3:</strong> Tank drains when \(y=0\): \(0 = 4 - kt \implies t = \dfrac{4}{k}\).</p>
<p><strong>Answer: (D)</strong> \(t = 4/k\).</p>
<p class='key-concept'>🔑 Key Concept: Torricelli's law — outflow rate ∝ √h. Always separate and integrate \(y^{-1/2}\,dy\).</p>
<p class='trap-warning'>⚠️ Trap: Writing \(dy/dt = ky\) (exponential decay) instead of \(-k\sqrt{y}\). The \(\sqrt{y}\) exponent changes the solution completely.</p></div>
Correct Answer: 4