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 :
z=f(x,y)=1250ln(yx^2)+45(y^2+x)(x^3 -2y)-(xy)^1/2
where
z= the weekly # of pounds of acetate fiber produced
x=the # of skilled workers at the plant
y= the # of unskilled workers at the plant
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
b)a week later, 20 skilled workers and 12 unskilled workers are employed due tonthe increasing demand of acetate fiber. what is the rate of change of output with respect to skilled worker?
Can you give me the answer. I have answer it almost in 1 week
a) To find the rate of change of the amount of acetate fiber produced, we need to calculate the partial derivative of z with respect to y (the number of unskilled workers), while holding x (the number of skilled workers) constant.
First, let's differentiate the given function, z, with respect to y:
∂z/∂y = ∂/∂y (1250ln(yx^2) + 45(y^2+x)(x^3 - 2y) - (xy)^1/2)
To do this, we'll need to apply the chain rule and the product rule for differentiation:
∂z/∂y = ∂/∂y (1250ln(yx^2)) + ∂/∂y (45(y^2+x)(x^3 - 2y)) - ∂/∂y ((xy)^1/2)
Differentiating each term:
∂z/∂y = 1250(1/y)ln(yx^2) + 45 * [2y(x^3 - 2y) + (y^2 + x) * (-2)] - (1/2) * (xy)^(-1/2)
Now, simplify the expression:
∂z/∂y = 1250ln(yx^2)/y + 90y(x^3 - 2y) - 2(y^2 + x) - (xy)^(-1/2)/2
This is the rate of change of the amount of acetate fiber produced with respect to the number of unskilled workers.
b) Similarly, to find the rate of change of output with respect to skilled workers (x), we need to calculate the partial derivative of z with respect to x, while holding y constant.
∂z/∂x = ∂/∂x (1250ln(yx^2) + 45(y^2+x)(x^3 - 2y) - (xy)^1/2)
Differentiating each term:
∂z/∂x = 1250(2x/y)ln(yx^2) + 45 * [(y^2 + x) * (3x^2) - 2y(x^3 - 2y)] - (1/2) * (xy)^(-1/2)
Simplifying the expression:
∂z/∂x = 2500xln(yx^2)/y + 45 * [(3x^3(y^2 + x)) - 2y(x^3 - 2y)] - (xy)^(-1/2)/2
This is the rate of change of output with respect to the number of skilled workers.