Use E q .(4) and Torricelli's Law [E q .(5)]. A tank has the...

Use $E q .(4)$ and Torricelli's Law $[E q .(5)]$. A tank has the shape of the parabola $y=x^{2}$, revolved around the $y$-axis. Water leaks from a hole of area $B=0.0005 \mathrm{m}^{2}$ at the bottom of the tank. Let $y(t)$ be the water level at time $t .$ How long does it take for the tank to empty if it is initially filled to height $y_{0}=1 \mathrm{m}$ .

Use $E q .(4)$ and Torricelli's Law $[E q .(5)]$.

A tank has the shape of the parabola $y=x^{2}$, revolved around the $y$-axis. Water leaks from a hole of area $B=0.0005 \mathrm{m}^{2}$ at the bottom of the tank. Let $y(t)$ be the water level at time $t .$ How long does it take for the tank to empty if it is initially filled to height $y_{0}=1 \mathrm{m}$ .

Key Concepts

Torricelli's Law

Torricelli's Law is a principle in fluid dynamics that uses Bernoulli's equation to relate the speed at which fluid exits an orifice to the height of the fluid above the orifice. Essentially, it states that the exit speed is proportional to the square root of the gravitational acceleration times the height of the fluid column, and it provides the foundational relationship for modeling fluid outflow from tanks.

Differential Equations in Fluid Dynamics

In problems involving fluid depletion, a differential equation is often derived that relates the rate of change of the fluid level to the properties of the tank and the fluid. This is achieved by equating the rate of volume change within the tank to the product of the efflux velocity and the area of the orifice, resulting in an equation that captures how the height of the fluid decreases over time.

Volume Calculation for Solids of Revolution

Determining the volume of a tank with a non-standard shape, like a parabolic profile rotated about an axis, involves integrating the cross-sectional areas along the axis of revolution. This process uses the method of disks or washers to account for the changing radius and thus the changing area at different heights, which is crucial for establishing the relationship between the fluid height and the contained volume.

Separation of Variables

Separation of variables is a common technique used to solve differential equations that arise in fluid dynamics problems. By rearranging the equation so that all terms involving the dependent variable (such as the fluid height) are on one side and all terms involving the independent variable (typically time) are on the other, one can integrate both sides to find an expression that describes the evolution of the fluid level over time.

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