Program Explanation Given information: The hemispherical tank is filled with water completely. The radius of the tank is 1 m and the radius of the bottom hole is 1 cm . Calculation: The given problem is of draining tank. The Toricelli’s law is used for a draining tank. The Toricelli’s law can be represented as, A ( y ) d y d t = − a 2 g y ........ (1) Here, A ( y ) is the horizontal cross sectional area of the tank and a is the area of the hole and the value of g = 32 ft/s 2 . In the given problem A ( y ) = A is constant for every height. The area of the hole a can be calculated as, a = π ( 1 100 ) 2 a = π 10000 The value of r ( y ) can be calculated with the help of right angle triangle made in hemisphere. r ( y ) = R 2 − y 2 = 1 − y 2 The horizontal cross sectional area of the tank A ( y ) = A can be calculated as, A = π ( 1 − y 2 ) Now substitute 32 for g , π 10 − 4 for a and π ( 1 − y 2 ) for A in equation (1) as, π ( 1 − y 2 ) d y d t = − π 10 − 4 2 ⋅ 9.75 y π ( 1 − y 2 ) d y d t = − π 10 − 4 19.5 y d y d t = − 1 10 − 4 19.5 1 ( y − 1 2 − y 3 2 ) Now use the method of separation of the variable in above differential equation. d y d t = − 1 10 − 4 19.5 1 ( y − 1 2 − y 3 2 ) ( y − 1 2 − y 3 2 ) d y = − 1 10 − 4 19.5 d t Now integrate both sides. ( y − 1 2 − y 3 2 ) d y = − 1 10 − 4 19

Differential Equations: Computing and Modeling (5th Edition), Edwards, Penney & Calvis

5th Edition

ISBN: 9780321816252

Author: C. Henry Edwards, David E. Penney, David Calvis

Publisher: PEARSON

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A naïve solution for the above-stated problem is to evaluate all of the possible combinations of different pieces of varying size to find the maximum revenue-generating combination. Although this method seems the easiest and quick to practice, it has exp…

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Chapter 1.4, Problem 62P

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Program Description: Purpose of the problem is to find the time when the tank goes to empty.

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Chapter 1.4, Problem 61P

Chapter 1.4, Problem 63P

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Chapter 1 Solutions

Differential Equations: Computing and Modeling (5th Edition), Edwards, Penney & Calvis

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Program Description: Purpose of the problem is to find the time when the tank goes to empty.

Solution Summary: The author explains how the Toricelli's law is used for a draining tank. The horizontal cross sectional area of the tank is constant for every height.

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Program Description: Purpose of the problem is to find the time when the tank goes to empty.

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Chapter 1 Solutions