The mass of the part of a metal rod that lies between its left end and a point [tex]$x$[/tex] meters to the right is [tex]$s = 4x^2$[/tex]. Find the linear density when [tex]$x$[/tex] is 3 m.
Answer (1)
Find the derivative of the mass function S = 4 x 2 with respect to x to obtain the linear density function: d x d S = 8 x .
Evaluate the linear density function at x = 3 : 8 ( 3 ) = 24 .
The linear density at x = 3 meters is 24 .
Explanation
Problem Analysis We are given the mass of a metal rod as a function of the distance x from its left end: S = 4 x 2 . We need to find the linear density of the rod at x = 3 meters. The linear density is the rate of change of mass with respect to length, which is the derivative of the mass function with respect to x .
Calculating the Derivative To find the linear density, we need to calculate the derivative of S with respect to x .
ρ = d x d S = d x d ( 4 x 2 ) Using the power rule, we have: d x d ( 4 x 2 ) = 4 ⋅ 2 x = 8 x So, the linear density ρ is given by ρ = 8 x .
Evaluating at x=3 Now, we need to find the linear density at x = 3 meters. We substitute x = 3 into the expression for ρ :
ρ ( 3 ) = 8 ( 3 ) = 24 Thus, the linear density at x = 3 meters is 24.
Final Answer The linear density of the metal rod at x = 3 m is 24.
Examples
Imagine you are designing a bridge and need to reinforce certain sections with metal rods. Knowing the linear density of the metal allows you to calculate the mass of a specific length of the rod. This is crucial for ensuring the structural integrity and weight distribution of the bridge. By understanding how mass varies with length, engineers can make informed decisions about material usage and placement, optimizing the bridge's design for safety and efficiency.
