Solution to Midterm Examination (International Business, WUCL)
Summer 2007 Instructor: 衛忠欣 (Jong-Shin Wei)
管理數學 (Mathematics for Management) (07)342-6031 ext.6201
Closed books/notes exam in 60 minutes; absolutely no talking nor borrowing items during exams. 可使用自己的字典或翻譯機。行動電話若響起，該生扣十分、以強調基本禮貌。09:30 am – 10:30 am, Wednesday, July 11, 2007. Watch your time and good luck!
務必原卷工整扼要作答、不得超出作答空間；如果字跡難以辨認，視同未答。畫蛇添足、答非所問、自曝其短，將不利得分。教師也有權公告具有特色之作答。Watch your time and have fun!
1. [20 points] Recall the example given in class, regarding three workers (A, B, and C) and a (labor-intensive) task to be completed. As before, assume perfect divisibility and fixed productivity. Furthermore, assume that
if A and B work together while C does not show up, it takes 20 days to complete it;
if A and C work together while B does not show up, it takes 20 days to complete it;
if B and C work together while A does not show up, it takes 40 days to complete it.
How many days does A need for the task if A works alone?
Ans: Without the loss of generosity, we assume that the amount of task (to be carried out) is k > 0. Each worker’s productivity is characterized by k divided by the number of days for this worker to contribute alone. Let x, y, and z represent the productivity of workers A, B, and C respectively.
k = 20x + 20y + 0z = 20x + 20y,
k = 20x + 20z,
k = 40y + 40z.
From 20x + 20y = 20x + 20z = k we have y = z, hence k = 40y + 40z leads to y = k/80, yielding y = (1/80)k.
It follows at once that x = (k - 20y)/20 = (3/80)k.
To conclude, A needs 80/3 days to complete the task if A works alone.
2. [20 points] Choose non-negative ordered pair (x, y) to maximize x + 2y
subject to x - y ≤ 4, x + y ≤ 8, x ≤ 2, and y ≤ 4. Must show key steps.
Ans: First we depict lines x - y = 4 and x + y = 8. Next, we identify closed half-spaces x - y ≤ 4, x + y ≤ 8, x ≤ 2, and y ≤ 4. The intersection of all these four half-spaces is the region with four vertex points (0, 0), (2, 0), (2, 4), and (0, 4). [In formal language, we shall say that constraints x - y ≤ 4 and x + y ≤ 8 are not binding.] We then examine the family of lines characterized by x + 2y = k. Visual inspection leads to (2, 4).
3. True or False. No explanation is necessary. 每答對一題，得4分。每答錯一題，倒扣1分。
(a) A line on the plane can not have more than two horizontal intercepts.
(b) A line on the plane must have at least one vertical intercept.
(c) A line on the plane must have the unique slope.
(d) Let a, b, c, d, e, and f be parameters (that are real numbers). Equations ax + by = c and dx + ey = f may have no solution in (x, y).
Ans: True. [兩條不相交的平行線]
(e) Behind the equation xy = 1 we can find two functions.
*Ans: True. [One function is f(x) := 1/x; another is g(x, y) := xy.]
4. [20 points] 在電影 A Beautiful Mind 裡，數學家 John Nash 的治學精神，與你(妳)的學習此門課，最大的不同為何？你(妳)的未來雇主，又將如何透過學校，了解此點呢？
5. [20 points] 下圖是一場五子棋目前賽況。你(妳)持白子待攻，對手持黑子。請小心以1, 2, 3, 4, …在圖上，分別依序標示接下來你(妳)與對手的棋路。當然，贏棋是你(妳)的目標。
Ans: (to be solved on July 12, 2007)