# Differential Equation : Torricelli's Law

Please help with the following problem. Provide step by step calculations for each.

(1) (a) Let h(t) and V (t) be the height and volume of water in a cylindrical tank at time t. If water leaks through a circular hole with area a at the bottom of the tank, Torricelli's law says that the rate of change of volume is given by the equation

dV

--- = - a*sqrt(2gh)

dt

where g is the acceleration due to gravity.

(i) Show that the height of the water decreases according to the equation

dh

-- = = - alpha*sqrt(h)

dt

for some constant alpha. What is alpha?

(ii) If we let h(t) = u(t)^2, find a differential equation for u.

(iii) Given that the tank is initially full and the height of the tank is 6 metres and the radius of the tank is 2 metres, how long does it take for all the water to drain from the tank, if the hole has radius 1cm?

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#### Solution Summary

This posting helps with calculation and analysis problems. Differential Equation and Torricelli's Law are investigated in the solution. Step by step calculations are given for each problem.