Union membership as a percentage of the labor force can be modeled by
M(x) = 0.0003x3 − 0.066x2 + 4.64x − 81, where x is the number of years after 1900,and M is membership as a percentage of the labor force.
a. Find the rate at which membership is changing in 1960.
b. Find the rate at which membership is changing in 1985.
M'(x) = .0009x^2 - .132x + 4.64
Now just evaluate M' for your years.
ty!
To find the rate at which membership is changing in a given year, we need to find the derivative of the membership function with respect to time. In this case, time is represented by x, the number of years after 1900.
Let's start with part (a) and find the rate of change of membership in 1960.
a. To find the rate of change in 1960, we need to substitute the value of x = 1960 into the derivative of the membership function.
First, let's find the derivative of the membership function, M'(x), by differentiating each term of the function:
M(x) = 0.0003x^3 − 0.066x^2 + 4.64x − 81
M'(x) = d/dx (0.0003x^3 − 0.066x^2 + 4.64x − 81)
M'(x) = 0.0009x^2 - 0.132x + 4.64
Now, substitute x = 1960 into the derivative function to find the rate of change in 1960:
M'(1960) = 0.0009(1960)^2 - 0.132(1960) + 4.64
Simplifying the equation will give us the rate of change in 1960.
b. Now, let's move on to part (b) and find the rate of change of membership in 1985.
Following the same steps as before, we need to substitute x = 1985 into the derivative of the membership function:
M'(1985) = 0.0009(1985)^2 - 0.132(1985) + 4.64
Simplifying this equation will give us the rate of change in 1985.
By solving the equations in parts (a) and (b), you will find the rate at which membership is changing in 1960 and 1985, respectively.