{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. 
a. Find the solution of the initial value problem.

Verified step by step guidance

1. Recognize that the differential equation given is \(h'(t) = -k \sqrt{h(t)}\), where \(h(t)\) represents the water depth at time \(t\), and \(k\) is a positive constant.
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 \(\int \frac{1}{\sqrt{h}} \, dh\) on the left and \(\int -k \, dt\) on the right. Recall that \(\int h^{-1/2} \, dh = 2 \sqrt{h}\).
4. After integration, you will have an equation involving \(\sqrt{h}\) and \(t\) plus an integration constant \(C\). Use the initial condition \(h(0) = H_m\) to solve for \(C\).
5. Finally, solve the resulting equation for \(h(t)\) explicitly to express the water depth as a function of time.

Key Concepts

Torricelli’s Law

Torricelli’s law describes the speed of fluid flowing out of an orifice under gravity, stating that the outflow velocity is proportional to the square root of the fluid’s height above the hole. In this problem, it leads to the differential equation h'(t) = -k√h, relating the rate of change of water depth to the current depth.

Separable Differential Equations

A separable differential equation can be written as a product of a function of t and a function of h, allowing variables to be separated on opposite sides of the equation. Here, h'(t) = -k√h can be rearranged to integrate both sides and find an explicit solution for h(t).

Initial Value Problem (IVP)

An initial value problem involves solving a differential equation with a given initial condition, such as h(0) = Hm. This condition allows determination of the constant of integration after solving the differential equation, yielding a unique solution that models the water depth over time.