{Use of Tech} Torricelli’s law An open cylindrical tank initially filled with water drains through a hole in the bottom of the tank according to Torricelli’s law (see figure). If h(t) is the depth of water in the tank for t≥0 s, then Torricelli’s law implies h′(t)=−k√h, where k is a constant that includes g=9.8m/s², the radius of the tank, and the radius of the drain. Assume the initial depth of the water is h(0)=Hm. 
b. Find the solution in k=0.1the case that and H=0.5m. 

Verified step by step guidance

1. Start with the given differential equation from Torricelli's law: \(h'(t) = -k \sqrt{h(t)}\), where \(k = 0.1\) and the initial condition is \(h(0) = H = 0.5\) meters.
2. Rewrite the differential equation in separable form: \(\frac{dh}{dt} = -k \sqrt{h}\) can be rearranged as \(\frac{dh}{\sqrt{h}} = -k \, dt\).
3. Integrate both sides: integrate \(\frac{1}{\sqrt{h}} \, dh\) on the left and integrate \(-k \, dt\) on the right. Remember to include the constant of integration after integrating.
4. Solve the integral on the left side, which is \(\int h^{-1/2} \, dh = 2 \sqrt{h}\), and the integral on the right side, which is \(-k t + C\) where \(C\) is the constant of integration.
5. Use the initial condition \(h(0) = 0.5\) to solve for the constant \(C\), then express \(h(t)\) explicitly as a function of \(t\) by isolating \(h\).

Key Concepts

Torricelli’s Law

Torricelli’s law describes the speed of fluid flowing out of an orifice under gravity, stating that the rate of change of the water height h(t) is proportional to the square root of h(t). Mathematically, it is expressed as h'(t) = -k√h, where k depends on gravity and the tank and hole dimensions.

Separable Differential Equations

The given differential equation h'(t) = -k√h is separable, meaning variables h and t can be separated on opposite sides of the equation. This allows integration with respect to each variable independently to find the explicit solution h(t).

Initial Value Problem (IVP)

An initial value problem involves solving a differential equation with a given initial condition, here h(0) = H. This condition is used to determine the constant of integration after solving the differential equation, ensuring the solution fits the physical scenario.