# The meaning of V ' ( t ) and H ' ( t ) , and whether these are positive, negative or zero. ### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805 ### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

#### Solutions

Chapter 2, Problem 12P

(a)

To determine

## To find:The meaning of V'(t) and H'(t) , and whether these are positive, negative or zero.

Expert Solution

V'(t) and H'(t) are positive.

### Explanation of Solution

Given information:

V(t) is the volume of the water in the tank and H(t) is the height of the water in the tank at time t .

Calculation:

The term V'(t) represents that the rate of change of the volume of the water in the tank at time t and represent H'(t) that the rate of change of the height of the water in the tank at time t .

In case when the water is flowing into the tank, the volume of the water V(t) in the tank will be increase as t increases.

And, H(t) of the water in the tank will be increase as t increases.

V(t) and H(t) are the increasing function, there derivatives V'(t) and H'(t) are positive.

Therefore, V'(t) and H'(t) are positive.

(b)

To determine

### To find:Whether the value of V''(t) is positive, negative or zero.

Expert Solution

The value of V''(t) is zero.

### Explanation of Solution

Given information:

V(t) is the volume of the water in the tank and H(t) is the height of the water in the tank at time t .

The water is flowing at a constant rate into the tank.

Calculation:

As given that, the water is flowing at a constant rate into the tank.

The amount of water increases at each time t is constant.

So, V'(t) is constant, and hence V''(t)=0 .

Therefore, the value of V''(t) is zero.

(c)

To determine

### To find:Whether the value of H''(t1),H''(t2),H''(t3) is positive, negative or zero.

Expert Solution

The value of H''(t1),H''(t2),H''(t3) are zero.

### Explanation of Solution

Given information:

V(t) is the volume of the water in the tank and H(t) is the height of the water in the tank at time t .

The water is flowing at a constant rate into the tank.

Calculation:

The tanker is one-quarter full at the time t1 .

So, the volume V(t1) at time t1 is constant, also the height H(t1) at time t1 is constant.

So, H'(t1)=0

And, H''(t1)=0

Similarly,

H''(t2)=0H''(t3)=0

Therefore, the value of H''(t1),H''(t2),H''(t3) are zero.

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