# The linear density at x = 1 m . ### 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 3.8, Problem 17E

(a)

To determine

## To find: The linear density at x=1 m.

Expert Solution

The linear density at x=1m is, ρ(1)=6kg/m.

### Explanation of Solution

Given:

The mass of the part of the metal rod which lies between its left end and a point x m away from the right end is 3x2kg.

Calculation:

The mass of the metal rod is, m(x)=3x2.

The linear density of the rod is the derivative of the mass and hence ρ(x)=6x.

The linear density at x=1m is ρ(1)=6kg/m.

(b)

To determine

### To find: The linear density at x=2 m.

Expert Solution

The linear density at x=2m is, ρ(2)=12kg/m.

### Explanation of Solution

From part (a), the linear density of the rod is, ρ(x)=6x.

The linear density at x=2m is,

ρ(2)=6(2)=12kg/m

The linear density at x=2m is, ρ(2)=12kg/m.

(c)

To determine

### To find: The linear density at x=3 m; and to tell the conclusion obtained from the parts (a), (b) and (c).

Expert Solution

The linear density at x=3m is, ρ(3)=18kg/m.

### Explanation of Solution

From part (a), the linear density of the rod is, ρ(x)=6x.

The linear density at x=3m is, ρ(3)=18kg/m.

Notice that the density function is an increasing function. Therefore, the density of the rod is highest at its right end and lowest at its left end.

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