Lesson One unusual mimimax problem on joint work
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<H2>One unusual mimimax problem on joint work</H2> <H3>Problem 1</H3>It takes Nicolai one and a half hours to paint the walls of a room and two hours to paint the ceiling. Elena needs exactly one hour to paint the walls of the same room and one hour to paint the ceiling. If Nicolai and Elena work together, what is the shortest possible time in minutes in which they can paint the walls and the ceiling of that room? <B>Solution</B> <pre> 1. Let us assign Nicolai to paint the walls, and let us assign Elena to paint the ceiling. After 1 hour working, Elena will complete painting ceiling. During this 1 hour Nicolai will complete painting of 2/3 of walls. Then Elena will switch for painting the walls with Nikolai. Their combined rate of work painting the walls is {{{1/90 + 1/60}}} = {{{2/180 + 3/180}}} = {{{5/180}}} = {{{1/36}}} of the walls surface per minute. Hence, it will take {{{((1/3))/((1/36))}}} = {{{36/3}}} = 12 minutes for them to complete the job painting the remaining {{{1/3}}} of the wall surface. Last calculation 60 minutes + 12 minutes = 72 minutes gives you the ANSWER. 2. In n.1 I simply "guessed" the right solution. Now I want to show you how to derive it MATHEMATICALLY. In the general case, let "w" and "c" be two real numbers between 0 and 1: 0 <= w, c <= 1. Let us assign Nicolai to paint w-th part of the wall area and c-th part of the ceiling. Accordingly, Elene will paint (1-w)-th part of the wall area and (1-c)-th part the ceiling. Our task is to chose the values w and c in a way to minimize the total time. The necessary condition for it is <U>THIS</U>: Nicolai and Elena should complete their personal assignments <U>SIMULTANEOUSLY</U>. (*) I will leave this statement without the proof (because it is self-evident). It is self-evident, but EXTREMELY helpful. 3. Now let see what does it mean in a quantitative sense. The time for Nicolai to complete his assignment is 90*w + 120*c minutes (90 is 90 minutes = 1.5 hours and 120 = 120 minutes = 2 hours). The time for Elena to complete her assignment is 60*(1-w) + 60*(1-c) minutes (60 is 60 minutes = 1 hour and 60 = 60 minutes = 1 hour). The condition (*) gives you an equation 90*w + 120*c = 60*(1-w) + 60*(1-c), which you can simplify to the form 5w + 6c = 4. (1) So, you see that the optimal solution must obey to the condition (1) to provide the minimal time. 4. Now, the time that Nicolai will spend for his personal assignment is N = 90w + 120c = 90w + 108c + 12c = 18*(5w + 6c) + 12c = 18*4 + 12c = 72 + 12c. And you see that N is always greater than 72, if c > 0. So, to provide minimal time 72 minutes for Nicolai, the value of "c" must be ZERO. Thus we get this solution w = {{{4/5}}}, c = {{{0}}}, which I "guessed" in n.1. 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