At an Oregon fiber-manufacturing facility, an analyst estimates that the weekly number of pounds of acetate fibers that can be produced is given by the function :
2 2 3 ( , ) 1250ln( ) 45( )( 2 ) z f x y yx y x x y xy
where the weekly # of pounds of acetate fiber produced the # of skilled workers at the plant the # of unskilled workers at the plant z x y
a) On the 2nd week of April, the production executive would like to see the effect of the number of unskilled worker towards the production performance. Therefore, the number of skilled workers is fixed and the number of unskilled worker to be employed at the production line is depending on the situation. Find the rate of change of the amount of acetate fiber produced in this week.
To find the rate of change of the amount of acetate fiber produced in the second week of April, we need to take the partial derivative of the function f(x, y, z) with respect to y.
Given the function: f(x, y, z) = 2x^2 + 3yln(z) + 45x(y - z) + yz
Taking the partial derivative with respect to y, we treat x and z as constants:
∂f/∂y = 0 + 3ln(z) + 45x - 45z + z
Simplifying further:
∂f/∂y = 3ln(z) + 45x - 44z + z
This calculates the rate of change of the amount of acetate fiber produced in the second week of April, considering the number of skilled workers is fixed and the number of unskilled workers is variable.