(a) Program Explanation Given information: The shape of the water tank is obtained by revolving the parabola x 2 = b y around y axis The vertical tank contains 4 ft water initially at 12 noon after one hour height of the water is dropped to 1 ft . 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 cone and k is constant. The cross sections of the tank are circles. The height is y 0 ft and the radius is x 0 ≥ 0 such that b y 0 = x 0 2 . Therefore, the radius of the tank can be calculated as, x 0 = b y 0 Therefore, the value of A ( y ) = A can be calculated as, A ( y 0 ) = π x 0 2 = π b y 0 Now define the function as, A ( y ) = π b y Now substitute π y 3 2 for A ( y ) in equation (1) as, π b y d y d t = − a 2 g y π y d y d t = − k y Therefore, the equation is as follows, π y d y d t = − k y ........ (2) Now use the method of separation of the variable to solve the equation (1). π y 1 2 d y d t = − k π y d y = − k d t Now integrate both sides. π ∫ y d y = ∫ − k d t π y 3 2 3 2 = − k t + c 2 3 π y 3 2 = − k t + c Therefore, the resultant equation is as follows, 2 3 π y 3 2 = − k t + c (b) Program Plan Intro Program Description: Purpose of the problem is to find the time when the tank goes to empty. (c) Program Plan Intro Program Description: Purpose of the problem is to find the radius of the circular hole.

5th Edition

ISBN: 9780321974235

Author: Calvis

Publisher: PEARSON CUSTOM PUB.(CONSIGNMENT)

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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…

Question

Chapter 1.4, Problem 59P

(a)

Program Plan Intro

Program Description: Purpose of the problem is to find the depth y(t) of remaining water

(b)

Program Plan Intro

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

(c)

Program Plan Intro

Program Description: Purpose of the problem is to find the radius of the circular hole.

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Chapter 1.4, Problem 58P

Chapter 1.4, Problem 60P

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Chapter 1 Solutions

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Program Description: Purpose of the problem is to find the depth y ( t ) of remaining water

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

Concept explainers

Program Description: Purpose of the problem is to find the depth y(t) of remaining water

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

Program Description: Purpose of the problem is to find the radius of the circular hole.

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Chapter 1 Solutions