#### To determine

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

#### Answer

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

#### Explanation

**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 3x2 kg.

**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=1 m is ρ(1)=6 kg/m.

#### To determine

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

#### Answer

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

#### Explanation

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

The linear density at x=2 m is,

ρ(2)=6(2)=12 kg/m

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

#### To determine

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

#### Answer

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

#### Explanation

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

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