   Chapter 2.7, Problem 17E

Chapter
Section
Textbook Problem

# The mass of the part of a metal rod that lies between its left end and a point x meters to the right is 3 x 2 kg. Find the linear density (see Example 2) when x is (a) 1 m, (b) 2 m, and(c) 3 m. Where is the density the highest? The lowest?

To determine

To find: The linear density when x is:

(a) 1m(b) 2m, and(c) 3m then decide where is the density highest & lowest?

Explanation

1) Concept:

The linear density of the rod is the derivative of mass (m) with respect to length (x).

2) Formula:

i. The linear density

ρ=  ddx (m)

ii. Power rule of derivative:

ddxxn=nxn-1

3) Given:

(a) x = 1m (b) x = 2m, and (c) x = 3m

Mass = m = 3x2 as shown in figure.

4) Calculation:

By using above formula, linear density

ρ=  ddx (m)

Substitute m = 3x2 and find derivative with respect to x.

Linear density

ρ=  ddx3x2

By using power rule of derivative,

Linear density

ρ=6x

(a)To find linear density at x = 1, substitute x = 1 in linear density ρ = 6x

Linear density

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