Hello! I'm Alex. I often get up at 6:30 am. Half an hour later(之后), I have my breakfast. At 7:30 a.m., I take a bus to school. We have Maths, Chinese and English classes from 8:00 a.m. to 11:30 a.m. I have lunch and talk with my classmates at school. We have three classes from 1:40 to 4:10 in the afternoon.

    After school, I often play with my classmates. We often play soccer, tennis and ping-gong. I go home at about 5:00 p.m. I usually have dinner at 6.00 p.m. Then I do my homework and watch TV. At about 10:30 p.m., I go to bed.

如图，已知点A（11，0），函数y=$\sqrt{x+1}$的图象上的动点P在x轴上的射影为H，且点H在点A的左侧，设|PH|=t，△APH的面积为f（t）

（1）求函数f（t）的解析式及t的取值范围．

（2）若a∈（0，2$\sqrt{3}$），求函数f（t）在（0，a上的最大值．