Question: (10 points each)

(a) Find the equation of the tangent line at (x, y) = (1, 1) when \({{x}^{2}}+{{y}^{3}}=1+x{{y}^{2}}\).

(b) The Cobb-Douglas model predicts that the monthly output q of a certain factory is given by

where k is the capital investment (in thousands of dollars) and w is the size of the labor force. Suppose that management wants to reduce its capital expenditures by increasing its labor force, keeping its output fixed at q = 12,000. In a certain month, k = 8 and the company is hiring workers a rate of 25 per month. How fast are the capital costs decreasing? Hint: This is a “related rates” problem.