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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621

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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621
Textbook Problem

Water is flowing at a constant rate into a spherical tank. Let V ( t ) be the volume of water in the tank and H ( t ) be the height of the water in the tank at time t .

(a) What are the meanings of V ' ( t ) and H ' ( t ) Are these derivatives positive, negative, or zero?

(b) Is V " ( t ) positive, negative, or zero? Explain.

(c) Let t 1 , t 2 and t 3 be the times when the tank is one-quarter full, half full, and three-quarters full, respectively. Are the values H " ( t 1 ) , H " ( t 2 ) , and H " ( t 3 ) positive, negative, or zero? Why?

To determine

a)

To find:

The meanings of V't and H'(t) and are these derivatives positive, negative or zero.

Explanation

Since water is flowing at a constant rate it means volume of water is increasing with a constant rate and this rate of change of volume is denoted by V't

Also V't is positive because volume of water is increasing with time until the spherical tank is full.

Therefore,

V't=dVdt

Similarly, H't is rate of change of height and it is also positive until the spherical tank is full because the height of water is increasing with time

To determine

b)

To explain:

Is V''t positive, negative or zero?

To determine

c)

To check:

The values of H''t1, H''t2,  and H''t3  are positive, negative or zero.

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