3. A conical container has a half-angle a as shown above. Liquid is poured in at a constant rate Q (volume per time). Simultaneously, liquid leaks out at a rate q that is proportional to the current height of the liquid, i.e. q = kh. (a) Set up the differential equation governing h(t). (b) Neglecting leakage (k = 0), solve for h(t) given that the container is initially empty. How long does it take to fill the container up to a total height H? Estimate the answer first using scaling arguments in addition to obtain an exact solution. (c) Now imagine that the container is full up to a height H at t= 0 but no more fluid is poured in (Q = 0). Solve for h(t). How long does it take for the container to be drained by the leak? Estimate the drain time using scaling arguments, then also obtain the solution exactly.

3. A conical container has a half-angle a as shown above. Liquid is poured in at a constant rate Q (volume per time). Simultaneously, liquid leaks out at a rate q that is proportional to the current height of the liquid, i.e. q = kh. (a) Set up the differential equation governing h(t). (b) Neglecting leakage (k = 0), solve for h(t) given that the container is initially empty. How long does it take to fill the container up to a total height H? Estimate the answer first using scaling arguments in addition to obtain an exact solution. (c) Now imagine that the container is full up to a height H at t= 0 but no more fluid is poured in (Q = 0). Solve for h(t). How long does it take for the container to be drained by the leak? Estimate the drain time using scaling arguments, then also obtain the solution exactly.

Elements Of Electromagnetics

7th Edition

ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

ChapterMA: Math Assessment

Section: Chapter Questions

Problem 1.1MA

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