Repeat Exercise 59 with the output function Q(x, y) = 1,731x + 925y + x²y − 2.7x² − 1.3y^3/2 and initial employment levels of x = 43 and y = 85.

Data from Exercises 59

It is estimated that the weekly output at a certain plant is given by Q(x, y) = 1,175x + 483y + 3.1x²y − 1.2x³ − 2.7y² units, where x is the number of skilled workers and y is the number of unskilled workers employed at the plant. Currently the workforce consists of 37 skilled and 71 unskilled workers.

a. Store the output function as Store 37 as X and 71 as Y and evaluate to obtain Q(37, 71). Repeat for Q(38, 71) and Q(37, 72).

b. Store the partial derivative Q_x(x, y) in your calculator, and evaluate Q_x(37, 71). Use the result to estimate the change in output resulting when the workforce is increased from 37 skilled workers to 38 and the unskilled workforce stays fixed at 71. Then compare with the actual change in output, given by the difference Q(38, 71) − Q(37, 71).

c. Use the partial derivative Q_y(x, y) to estimate the change in output that results when the number of unskilled workers is increased from 71 to 72 while the number of skilled workers stays at 37. Compare with the actual change Q(37, 72) − Q(37, 71).