A spherical tank of radius R is half-filled with water....

A spherical tank of radius $R$ is half-filled with water. Suppose that water leaks through a hole in the bottom of area $B$ . Let $y(t)$ be the water level at time $t($ seconds $) .$ (a) Show that $\frac{d y}{d t}=\frac{-8 B \sqrt{y}}{\pi\left(2 R y-y^{2}\right)}$. (b) Show that for some constant $C$, $$\frac{\pi}{60 B}\left(10 R y^{3 / 2}-3 y^{5 / 2}\right)=C-t$$ (c) Use the initial condition $y(0)=R$ to compute $C,$ and show that $C=t_{e},$ the time at which the tank is empty. (d) Show that $t_{e}$ is proportional to $R^{5 / 2}$ and inversely proportional to $B .$

A spherical tank of radius $R$ is half-filled with water. Suppose that water leaks through a hole in the bottom of area $B$ . Let $y(t)$ be the water level at time $t($ seconds $) .$
(a) Show that $\frac{d y}{d t}=\frac{-8 B \sqrt{y}}{\pi\left(2 R y-y^{2}\right)}$.
(b) Show that for some constant $C$,
$$\frac{\pi}{60 B}\left(10 R y^{3 / 2}-3 y^{5 / 2}\right)=C-t$$
(c) Use the initial condition $y(0)=R$ to compute $C,$ and show that $C=t_{e},$ the time at which the tank is empty.
(d) Show that $t_{e}$ is proportional to $R^{5 / 2}$ and inversely proportional to $B .$

Key Concepts

Dimensional Analysis and Scaling Laws

This concept is used to determine how various physical quantities scale with one another. By analyzing the units and dependencies, one can deduce relationships such as the proportionality of the emptying time to the tank's radius raised to a specific power or its inverse proportionality with the area of the hole, without necessarily performing a full derivation.

Geometric Analysis of Spherical Tanks

This involves using the geometric properties of a sphere to relate the water level to the volume of water contained in the tank. Understanding how the cross-sectional area changes with height is crucial for determining the rate at which water drains, as it directly affects the relationship between the water level and the leakage rate.

Differential Equation Modeling

This concept involves formulating the physical process as a differential equation, which describes how a quantity changes over time. In the context of draining a tank, the rate of change of the water level is modeled by a differential equation that relates the water height and the geometric properties of the tank to the outflow rate.

Separation of Variables

This mathematical technique is used to solve differential equations by rewriting them so that each variable and its differential appear on opposite sides of the equation. It allows the integration of each side with respect to its own variable, leading to a solution that relates the dependent and independent variables.

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