Find the linear density of the wire when x=4m.

[The legend]

Homework Statement

The mass of part of a wire is x(1+sqrt(x)) kilograms, where x
is measured in meters from one end of the wire. Find the
linear density of the wire when x=4m.



2. The attempt at a solution

Actually what i did was put in x = 4 in the mass equation given and divide by x since linear density = mass/length. And i get the answer to be 3.

But i don't know whether that's right or not.
Any help appreciated.

 

[Hootenanny]

Homework Statement

The mass of part of a wire is x(1+sqrt(x)) kilograms, where x
is measured in meters from one end of the wire. Find the
linear density of the wire when x=4m.



2. The attempt at a solution

Actually what i did was put in x = 4 in the mass equation given and divide by x since linear density = mass/length. And i get the answer to be 3.

But i don't know whether that's right or not.
Any help appreciated.

The way I interpret the question is the expression x(1+sqrt(x)), gives you the mass at a point x, i.e. that at x=1 the mass is 2, at x=2 the mass is 2+2sqrt(2), etc. If that is the case, then your method wouldn't be correct. Instead, to get the total mass of the wire, what you need to do as sum over the contribution of each point on the wire. Can you think of any method that will do that for you?

 

[The legend]

i would say it is integration that will help, but i haven't been taught that yet.

 

[Hootenanny]

i would say it is integration that will help, but i haven't been taught that yet.

Perhaps I am misinterpreting the question then.

 

[tiny-tim]

Hi Hoot! Hi The legend!

(have a square-root: √ )

The mass of part of a wire is x(1+sqrt(x)) kilograms, where x
is measured in meters from one end of the wire. Find the linear density of the wire when x=4m.

Isn't it differentiation?

I think they're saying that the total mass from 0 to x is x(1 + √x), and they want the density for a tiny bit round x = 4.

 

[The legend]

Perhaps I am misinterpreting the question then.

I took this sum from Stewart's book for calculus (Early Transcendentals 6th edition) and this sum was found under the chapter on derivatives(differentiation)...just to say

so how do I approach it then?

 

[Hootenanny]

I took this sum from Stewart's book for calculus (Early Transcendentals 6th edition) and this sum was found under the chapter on derivatives(differentiation)...just to say

so how do I approach it then?

If this is under differentiation, then I would suggest that tiny-tim has the correct interpretation. That is, the expression defines the mass of the segment of wire between 0 and x. In which case, the density is simply the rate of change of mass with respect to length.

P.S. Hi tim!

 

[tiny-tim]

Hi Hoot!

How's it goin'? ​

 

[The legend]

Hi Hoot! Hi The legend!

(have a square-root: √ )Isn't it differentiation?

I think they're saying that the total mass from 0 to x is x(1 + √x), and they want the density for a tiny bit round x = 4.

Hello tiny-tim!

But the question says that it is for a part of the wire.

The "mass of part" of a wire is x(1+sqrt(x)) kilograms,

But i did try it taking your way.
So what i did was this:

i differentiated the given mass function w.r.t length which gives

$$\frac{dm}{dl} = 1 + \frac{3 \sqrt(x)}{2}$$
this is the linear density at any tiny little place. So what i did is I put in x=4 in the equation and got the answer to be 4 kg/m³

 

[The legend]

...that took some editing there^...(latex)

is it right?

 

[tiny-tim]

Yup! 1 + 3√4/2 = 4 looks fine.

 

[The legend]

thanks a lot!