Question

4£®Èçͼ£¬O£¬P£¬QÈýµØÓÐÖ±µÀÏàͨ£¬OP=3ǧÃ×£¬PQ=4ǧÃ×£¬OQ=5ǧÃ×£¬Ïּס¢ÒÒÁ½¾¯Ô±Í¬Ê±´ÓOµØ³ö·¢ÔÈËÙÇ°ÍùQµØ£¬¾­¹ýtСʱ£¬ËûÃÇÖ®¼äµÄ¾àÀëΪf£¨t£©£¨µ¥Î»£ºÇ§Ã×£©£®¼×µÄ·ÏßÊÇOQ£¬ËÙ¶ÈΪ5ǧÃ×/Сʱ£¬ÒҵķÏßÊÇOPQ£¬ËÙ¶ÈΪ8ǧÃ×/Сʱ£¬ÒÒµ½´ïQµØºóÔÚÔ­µØµÈ´ý£®Éèt=t₁ʱÒÒµ½´ïPµØ£¬t=t₂ʱÒÒµ½´ïQµØ£®
£¨1£©Çót₁Óëf£¨t₁£©µÄÖµ£»
£¨2£©ÒÑÖª¾¯Ô±µÄ¶Ô½²»úµÄÓÐЧͨ»°¾àÀëÊÇ3ǧÃ×£¬µ±t₁¡Üt¡Üt₂ʱ£¬Çóf£¨t£©µÄ±í´ïʽ£¬²¢ÅжÏf£¨t£©ÔÚ[t₁£¬t₂]ÉϵÄ×î´óÖµÊÇ·ñ³¬¹ý3£¿ËµÃ÷ÀíÓÉ£®

Answer 1

·ÖÎö £¨1£©ÓÃOP³¤¶È³ýÒÔÒÒµÄËٶȼ´¿ÉÇóµÃt₁=$\frac{3}{8}$£¬µ±ÒÒµ½´ïPµãʱ£¬¿ÉÉè¼×µ½´ïAµã£¬Á¬½ÓAP£¬·ÅÔÚ¡÷AOPÖиù¾ÝÓàÏÒ¶¨Àí¼´¿ÉÇóµÃAP£¬Ò²¾ÍµÃ³öf£¨t₁£©£»
£¨2£©Çó³öt₂=$\frac{7}{8}$£¬Éèt$¡Ê[\frac{3}{8}£¬\frac{7}{8}]$£¬ÇÒtСʱºó¼×µ½´ïBµØ£¬¶øÒÒµ½´ïCµØ£¬²¢Á¬½ÓBC£¬Äܹ»ÓÃt±íʾ³öBQ£¬CQ£¬²¢ÇÒÖªµÀcos$¡ÏOQP=\frac{4}{5}$£¬ÕâÑù¸ù¾ÝÓàÏÒ¶¨Àí¼´¿ÉÇó³öBC£¬¼´f£¨t£©£¬È»ºóÇó¸Ãº¯ÊýµÄ×î´óÖµ£¬¿´ÊÇ·ñ³¬¹ý3¼´¿É£®

½â´ð ½â£º£¨1£©¸ù¾ÝÌõ¼þÖª${t}_{1}=\frac{3}{8}$£¬Éè´Ëʱ¼×µ½´ïAµã£¬²¢Á¬½ÓAP£¬ÈçͼËùʾ£¬ÔòOA=$5¡Á\frac{3}{8}=\frac{15}{8}$£»
¡àÔÚ¡÷OAPÖÐÓÉÓàÏÒ¶¨ÀíµÃ£¬f£¨t₁£©=AP=$\sqrt{O{A}^{2}+O{P}^{2}-2OA•OP•cos¡ÏAOP}$=$\sqrt{£¨\frac{15}{8}£©^{2}+9-\frac{45}{4}•\frac{3}{5}}=\frac{3\sqrt{41}}{8}$£¨Ç§Ã×£©£»
£¨2£©¿ÉÒÔÇóµÃ${t}_{2}=\frac{7}{8}$£¬ÉètСʱºó£¬ÇÒ$\frac{3}{8}¡Üt¡Ü\frac{7}{8}$£¬¼×µ½´ïÁËBµã£¬ÒÒµ½´ïÁËCµã£¬ÈçͼËùʾ£º
ÔòBQ=5-5t£¬CQ=7-8t£»
¡àÔÚ¡÷BCQÖÐÓÉÓàÏÒ¶¨ÀíµÃ£¬f£¨t£©=BC=$\sqrt{£¨5-5t£©^{2}+£¨7-8t£©^{2}-2£¨5-5t£©£¨7-8t£©•\frac{4}{5}}$=$\sqrt{25{t}^{2}-42t+18}$£»
¼´f£¨t£©=$\sqrt{25{t}^{2}-42t+18}$£¬$\frac{3}{8}¡Üt¡Ü\frac{7}{8}$£»
Éèg£¨t£©=25t²-42t+18£¬$\frac{3}{8}¡Üt¡Ü\frac{7}{8}$£¬g£¨t£©µÄ¶Ô³ÆÖáΪt=$\frac{21}{25}$$¡Ê[\frac{3}{8}£¬\frac{7}{8}]$£»
ÇÒ$g£¨\frac{3}{8}£©=\frac{369}{64}£¬g£¨\frac{7}{8}£©=\frac{25}{64}$£»
¼´g£¨t£©µÄ×î´óֵΪ$\frac{369}{64}$£¬Ôò´Ëʱf£¨t£©È¡×î´óÖµ$\frac{3\sqrt{41}}{8}£¼3$£»
¼´f£¨t£©ÔÚ[t₁£¬t₂]ÉϵÄ×î´óÖµ²»³¬¹ý3£®

µãÆÀ ¿¼²éÓàÏÒ¶¨ÀíµÄÓ¦Óã¬ÒÔ¼°¶þ´Îº¯ÊýÔÚ±ÕÇø¼äÉÏ×îÖµµÄÇ󷨣®