The mass of the part of a metal rod that lies between its left end and a point x meters to the right is 5x2 kg. (See this example.) (a) Find the linear density when x is 1 m. kg/m (b) Find the linear density when x is 4 m. kg/m (c) Find the linear density when x is 5 m. kg/m Where is the density the highest? At the right end of the rod. At the left end of the rod. In the middle of the rod. Where is the density the lowest? At the right end of the rod. At the left end of the rod. In the middle of the rod. Need Help? Read It Watch It

a) 10 kg/m

b) 40 kg/m

c) 50 kg/m

Density will be highest at right end of the rod and lowest at left end of the rod.

Explanation:

Formula to calculate linear density is the derivative of mass with respect to x (position).

Linear Density = dm/dx

So, here we have mass = m = 5. In order to calculate linear density just take the derivative of this mass function with respect to x and then plug in the value of x in that derivative.

Solution:

1. m = 5 , x = 1

Linear Density = dm/dx = d5/dx = (5*2)x = 10x

Linear Density = 10x

Now, plug in the value of x in this function to get the numerical value of linear density.

Linear Density = 10*(1) = 10 kg/m

2. when x = 4,

Similarly, plug the value of x in the obtained function

Linear Density = 10x = 10 (4) = 40 kg/m

3. When x = 5,

Linear Density = 10x = 10*(5) = 50 kg/m.

Density will be highest at right end of the rod and lowest at left end of the rod as seen from the equation of linear density that it is proportional to x which is position and it is increasing to the right and decreasing to the left. Hence, density will be highest on right end and lowest on left end.

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