HMMT 二月 2000 · CALC 赛 · 第 10 题
HMMT February 2000 — CALC Round — Problem 10
题目详情
- A mirror is onstru ted in the shap e of y equals p x for 0 x 1, and 1 for 1 < x < 9. A ra y of ligh t en ters at (10,1) with slop e 1. Ho w man y times do es it b oun e b efore lea ving?
解析
- Solution: Throughout this solution w e will use the fa t that when ligh t b oun es o a mirror, the angle of in iden e is equal to the angle of re e tion. First the b eam hits the p oin t (8,-1), then (6,1), (4,-1), (2,1), and then is tra v elling along the line y = x 1. p 1 p 5 1 p 5 Th us the b eam hits the parab ola at the p oin t (1 + ; ). T o estimate 5, noti e 2 2 p p 500 1 p 5 2 2 that 22 = 484 and 23 = 529, so 5 = = 2 : 2 : : : . Th us = : 6 : : : , so the ligh t 10 2 hits the parab ola at appro ximately (.4,-.6). The slop e of the tangen t to the parab ola 1 1 = 2 at this p oin t is ( : 4) , whi h is ab out -.8, so w e need to nd the slop e of the b eam 2 after it re e ts o of this tangen t. F or purp oses of nding this slop e, hange o ordinates so that the p oin t of in terse tion is the origin. The b eam is oming in along y = x , and y = 1 : 2 x is p erp endi ular to the tangen t. The diagram b elo w should larify the setup. perpendicular to tangent reflected beam y=x tangent W e will nd the new path of the ligh t b y nding the re e tion ab out the line y = 1 : 2 x of a p oin t on its in oming path. W e kno w the p oin t (1,1.2) is on the line y = 1 : 2 x , so a p erp endi ular through this p oin t is y 1 : 2 = : 8( x 1), whi h in terse ts y = x at the p oin t (1.1,1.1). Th us the new path go es through the p oin t (.9,1.3), so it has slop e 1.4 (all v alues rounded to one de imal pla e). Going ba k to our original o ordinate system, the ligh t is no w tra v elling along the line y + : 6 = 1 : 4( x : 4), so it next hits the mirror at (1.5,1). After that the x o ordinate in reases b y 2 = 1 : 4 = 1 : 4 b et w een b oun es, so it hits (2.9,-1), (4.3,1), (5.7,-1), (7.1,1), (8.5,-1), and nally (9.9,1). A loser examination of the appro ximations made (e.g. b y re ning them to t w o de imal pla es) rev eals that the last b oun e is a tually further to the left (at (9.21,1), to b e more pre ise), so indeed the ligh t do es b oun e 12 times.