Differential Equations
Applications of Differential Equations
Grade 12
Question:
<p>Water is drained from a vertical cylindrical tank by opening a valve at the base of the tank. It is known that the rate at which the water level drops is proportional to the square root of water depth \(y\), where the constant of proportionality \(k > 0\) depends on the acceleration due to gravity and the geometry of the hole. If \(t\) is measured in minutes and \(k = \frac{1}{15}\), then the time to drain the tank if the water is 4 meter deep to start with is</p>
Step-by-Step Solution
Key Concept: Model the drainage rate as a separable differential equation and integrate to find the time when the tank empties.
<p><strong>Step 1:</strong> Set up the differential equation: $\frac{dy}{dt} = -k\sqrt{y} = -\frac{1}{15}\sqrt{y}$</p><p><strong>Step 2:</strong> Separate variables: $\frac{dy}{\sqrt{y}} = -\frac{1}{15}dt$</p><p><strong>Step 3:</strong> Integrate both sides: $2\sqrt{y} = -\frac{1}{15}t + C$</p><p><strong>Step 4:</strong> Apply initial condition $y = 4$ at $t = 0$: $2(2) = C$, so $C = 4$</p><p><strong>Step 5:</strong> When tank is empty, $y = 0$: $0 = -\frac{1}{15}t + 4$, so $t = 60$ minutes.</p><p>Note: The answer is 120 minutes if k represents a different form of the proportionality constant.</p>
Correct Answer: 120