Question

The mass of the part of a metal rod that lies between its left end and a point \(x\) meters to the right is \(3{x^2}{\rm{ }}kg\). 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?

Short answer

(a). \({\rm{6 kg/m}}\)

(b). \({\rm{12 kg/m}}\)

(c). \({\rm{18 kg/m}}\)

Step-by-step solution

Linear Density.

The Linear Density can be defined as the ratio between the mass of the object and the length of that object where the mass of the unit length is being considered at a time.

Deriving function for the linear density of the rod:

(a)

The given function for the mass is:

\(M = 3{x^2}\)

Then, the linear density is given by:

\(\begin{aligned}\rho &= \frac{{dM}}{{dx}}\\ &= \frac{d}{{dx}}\left( {3{x^2}} \right)\\ &= 6x\end{aligned}\)

Now, the linear density for 1 metre length:

\(\rho \left( 1 \right) = 6\left( 1 \right) = 6{\rm{ kg/m}}\)

Calculating the linear density for 2 m length:

(b)

The linear density for 2 metre length:

\(\rho \left( 2 \right) = 6\left( 2 \right) = 12{\rm{ kg/m}}\)

Calculating the linear density for 3 m length:

(c)

The linear density for 3 metre length:

\(\rho \left( 3 \right) = 6\left( 3 \right) = 18{\rm{ kg/m}}\)

Hence,the highest density is at the right end and lowest at the left end of the rod.