A company assembles gadgets. It can make each gadget either by hand or with a special gadget-making machine. Each gadget can be assembled in 15 minutes by a worker or in 5 minutes by the machine. The firm can also assemble some of the gadgets by hand and some with machines. Both types of work are perfect substitutes, and they are the only inputs necessary to produce the gadgets. It costs the firm $30 per hour to use the machine and $10 per hour to hire a worker. The firm wants to produce 120 gadgets.

a) What are the cost-minimizing input quantities? Illustrate your answer with a clearly labeled graph.

b) What are the cost-minimizing input quantities if it costs the firm $20 per hour to use the machine, and $10 per hour to hire a worker? Illustrate your answer with a graph.

c) Write down the equation of the firm's production function for the firm. Let G be the number of gadgets assembled, M the number of hours the machines are used, and L the number of hours of labor.