Suppose there are two types of plumbers: competent and incompetent. Both types of plumbers can get certified, but the certification takes more time and effort for the incompetent types. Competent ones have to spend C months preparing for the certification exam; incompetent ones take twice as long. Certified plumbers can earn 100 (thousand dollars) each year working on building sites for licensed contractors. Uncertified plumbers can only earn 25 (thousand dollars) each year in freelance work; licensed contractors won't hire them. Each type of plumber gets a payoff equal toSM, where S is the salary measured in thousands of dollars and M is the number of months spent getting certified.

What is the lowestvalue of C for which a competent plumber will choose to signal with this certification but an incompetent one will not? (hint: find the entire range first).