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: \[ z=f(x, y)=14500 x+3000 y+5 x^{2} y-7 x^{3} \] Where: \( z= \) the weekly \# of pounds of acetate fiber \( x= \) the \# of skilled workers at the plant \( y= \) the \# of unskilled workers at the plant Determine the following: A) The weekly number of pounds of fiber that can be produced with 13 skilled workers and 32 unskilled workers. Answer \( =\square \) B) Find an expression \( \left(f_{x}\right) \) for the rate of change of output with respect to the number of skilled workers. Answer \( =f_{x}=\square \) C) Find an expression \( \left(f_{y}\right) \) for the rate of change of output with respect to the number of unskilled workers. Answer \( =f_{y}=\square \) D) Find the rate of change of output with respect to skilled workers when 13 skilled workers and 32 unskilled workers are employed. (Your answer will be a number.) Answer \( =\square \) Anskilled workers are employed. (Your answer will be a number.) Answer \( =\square \)

To tackle this fiber-manufacturing problem, let’s roll up our sleeves and dive into the math! **A)** To find the weekly number of pounds of fiber produced with 13 skilled workers and 32 unskilled workers, we substitute \( x = 13 \) and \( y = 32 \) into the function: \[ z = f(13, 32) = 14500(13) + 3000(32) + 5(13)^2(32) - 7(13)^3 \] Calculating gives us: \[ z = 188500 + 96000 + 5(169)(32) - 7(2197) \] \[ z = 188500 + 96000 + 27040 - 15379 \] \[ z = 188500 + 96000 + 27040 - 15379 = 285161 \] So, the weekly number of pounds of fiber produced is: **Answer \( = 285161 \)** --- **B)** The expression for the rate of change of output with respect to the number of skilled workers, \( f_{x} \), is found by taking the partial derivative with respect to \( x \): \[ f_{x} = \frac{\partial}{\partial x}(14500x + 3000y + 5x^2y - 7x^3) \] Calculating this gives: \[ f_{x} = 14500 + 10xy - 21x^2 \] So, the expression for \( f_{x} \) is: **Answer \( = f_{x} = 14500 + 10xy - 21x^2 \)** --- **C)** Similarly, for the rate of change of output with respect to the number of unskilled workers, \( f_{y} \): \[ f_{y} = \frac{\partial}{\partial y}(14500x + 3000y + 5x^2y - 7x^3) \] Calculating gives: \[ f_{y} = 3000 + 5x^2 \] So, the expression for \( f_{y} \) is: **Answer \( = f_{y} = 3000 + 5x^2 \)** --- **D)** Now to find the rate of change of output with respect to skilled workers when \( x = 13 \) and \( y = 32 \), substitute these values into the expression for \( f_{x} \): \[ f_{x} = 14500 + 10(13)(32) - 21(13)^2 \] Calculating gives: \[ f_{x} = 14500 + 4160 - 21(169) \] \[ f_{x} = 14500 + 4160 - 3549 = 17611 \] Thus, the rate of change of output with respect to skilled workers under the given conditions is: **Answer \( = 17611 \)**