A soap manufacturer has the production function q = LK 2 and faces a wage rate of $45 and rental rate of capital of $90. (a) Find the cost-minimizing ratio of capital to labor. (This is also called the output expansion path). (b) Suppose the firm plans to produce q=1,000. Determine the cost-minimizing amount of labor and capital. (c) How much will it cost to produce q=1,000 (long run)? (d) Suppose the firm decides to increase its output to q = 1331, but is unable to increase its quantity of capital from the amount in b. How much labor will the firm need to employ and what will be the cost of producing q = 1331? (e) If the firm continues to produce q = 1331 over a long period of time, how will it alter its capital and labor from part d.? What will it cost to produce q = 1331 in the long run? (f) On a graph, draw the two isoquants from above and the isocost lines corresponding to the LR and SR costs (there should be three). Identify the optimal L and K for each part on the graph.