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Unformatted text preview: MAT-235, 2008-2009 Problem Set 4 Solutions 1. There is nothing more annoying than studying math, so Peter is trying to minimize his study time for passing MAT-235. The overall mark will be composed by 15% from the problem sets, 45% from the term tests and 40% from the final exam. Peter expects his mark P for the problem sets to be proportional to the time x he spends on them (lets say, P = x ), his mark for the term tests to be T = 2 x 3 + y , where y is the time he spends studying specifically for the term tests and his mark F for the final to be F = x 2 + y 2 + z , where z is the time he studies specifically for the final. What is the minimum amount of time for achieving the pass mark of 50 points, and how should he break down this time? (The above model is not realistic, because in reality the marks P,T and F cannot go higher than 100 each. Comment on how this would influence your answer.) Solution: Clearly, the least amount of time will go with the lowest pass score of 50, therefore we need to minimize the function: f ( x,y,z ) = x + y + z subject to the constraint: g ( x,y,z ) = 0 . 15 x + 0 . 45 2 x 3 + y + 0 . 40 x 2 + y 2 + z = 50 . If we try to apply the method of Lagrange multipliers, we will see that f = (1 , 1 , 1) and g = (0 . 65 , . 65 , . 4) , so f and g are never proportional. (Geometrically: Both the level curves of f and the constraint g = 50 are planes, which however, are not parallel.) Therefore, there are no critical problems for the problem with constraints. However, the problem has a boundary since all x,y,z are , and the minimum should appear at the boundary. We write: f ( x,y,z ) = x + y + z = 1 . 65 (0 . 65 x + 0 . 65 y + 0 . 40 z ) + . 25 . 65 z = 1 . 65 g ( x,y,z ) + . 25 . 65 z. Since g ( x,y,z ) = 50 and z , this is 1 . 65 50 = 76 . 9 , and the minimum value of 76 . 9 is achieved when z = 0 . Therefore, Peter should not study at all for the final exam, and should break down a time of 76 . 9 between problem sets and term tests, in any way he wants. The restriction that the marks cannot exceed 100 would not influence the answer, since with 76 . 9 time units of work none of the marks P,T or F can go above 100 , no matter how Peter breaks up this time....
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