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
b) A week later, 20 skilled workers and 12 unskilled workers are employed due to the increasing demand of acetate fiber. What is the rate of change of output with respect to skilled worker?
To find the rate of change of output with respect to skilled workers, we need to calculate the partial derivative of the function z=f(x,y) with respect to x, denoted as ∂z/∂x.
Given the function z = f(x, y) = 1250ln(yx^2) + 45(y^2 + x)(x^3 - 2y) - √(xy), we can differentiate it with respect to x while treating y as a constant.
∂z/∂x = ∂/∂x (1250ln(yx^2) + 45(y^2 + x)(x^3 - 2y) - √(xy))
To differentiate each term individually, we can consider them as separate functions and apply the rules of differentiation.
∂/∂x (1250ln(yx^2)) = 0 (since the derivative of a constant multiplied by a function ln(yx^2) is zero)
∂/∂x (45(y^2 + x)(x^3 - 2y))
= 45 * (∂/∂x (y^2 + x))(x^3 - 2y) + (y^2 + x) * (∂/∂x(x^3 - 2y))
= 45 * (0 + 1)(x^3 - 2y) + (y^2 + x) * (3x^2 - 0)
= 45(x^3 - 2y) + (y^2 + x)(3x^2)
∂/∂x (√(xy)) = ∂/∂x (x^(1/2)y^(1/2))
= (1/2)x^(-1/2)y^(1/2)
Now, combining all these partial derivatives, we get:
∂z/∂x = 0 + 45(x^3 - 2y) + (y^2 + x)(3x^2) + (1/2)x^(-1/2)y^(1/2)
Simplifying further:
∂z/∂x = 45x^3 - 90y + 3xy^2 + 3x^3y + (1/2)x^(-1/2)y^(1/2)
To find the rate of change of output with respect to skilled workers, we substitute the given values of x and y (20 skilled workers and 12 unskilled workers) into this equation:
∂z/∂x = 45(20)^3 - 90(12) + 3(20)(12)^2 + 3(20)^3(12) + (1/2)(20)^(-1/2)(12)^(1/2)
Calculating this expression will give you the rate of change of output with respect to skilled workers.
To find the rate of change of output with respect to skilled workers, we need to calculate the partial derivative of the function f(x, y) with respect to x.
Step 1: Take the partial derivative of the first term of the function with respect to x. Since y and x^2 are independent of x, the derivative of 1250ln(yx^2) with respect to x is 0.
Step 2: Take the partial derivative of the second term of the function with respect to x. Using the product rule, we differentiate each term and multiply it by the other term, then sum them up. The derivative of 45(y^2+x) with respect to x is 45 * (x^3 - 2y).
Step 3: Take the partial derivative of the third term of the function with respect to x. Using the chain rule, we differentiate (xy)^1/2 with respect to (xy) and multiply it by the derivative of xy with respect to x, which is y. This gives us (1/2)(xy)^(-1/2) * y = (1/2)(xy)^(-1/2) * y.
Step 4: Combine the derivatives obtained in steps 1, 2, and 3. The partial derivative of f(x, y) with respect to x is:
0 + 45 * (x^3 - 2y) + (1/2)(xy)^(-1/2) * y = 45x^3 - 90y + (1/2)(xy)^(-1/2) * y
Step 5: Evaluate the derivative at the given values of x and y. Since there are now 20 skilled workers and 12 unskilled workers, we substitute x = 20 and y = 12 into the derived expression.
Rate of change of the output with respect to skilled worker = 45(20)^3 - 90(12) + (1/2)(20*12)^(-1/2) * 12 = 45(8000) - 90(12) + (1/2)(240)^(-1/2) * 12 = 360000 - 1080 + 1.732 * 12 = 361932.196