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# Differential Equation : Torricelli's Law

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(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?

##### 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.

Solution provided by:
###### Education
• BSc , Wuhan Univ. China
• MA, Shandong Univ.
###### Recent Feedback
• "Your solution, looks excellent. I recognize things from previous chapters. I have seen the standard deviation formula you used to get 5.154. I do understand the Central Limit Theorem needs the sample size (n) to be greater than 30, we have 100. I do understand the sample mean(s) of the population will follow a normal distribution, and that CLT states the sample mean of population is the population (mean), we have 143.74. But when and WHY do we use the standard deviation formula where you got 5.154. WHEN & Why use standard deviation of the sample mean. I don't understand, why don't we simply use the "100" I understand that standard deviation is the square root of variance. I do understand that the variance is the square of the differences of each sample data value minus the mean. But somehow, why not use 100, why use standard deviation of sample mean? Please help explain."
• "excellent work"
• "Thank you so much for all of your help!!! I will be posting another assignment. Please let me know (once posted), if the credits I'm offering is enough or you ! Thanks again!"
• "Thank you"
• "Thank you very much for your valuable time and assistance!"

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