(1)先证明AC1⊥A1C,再证明AB⊥平面AA1C1C,可得AB⊥AC1,利用线面垂直的判定定理,可得结论;
(2)确定点P到平面BB1C1C的距离等于点A到平面BB1C1C的距离,表示出三棱锥P-BCC1的体积,利用导数方法求最值.
(1)证明:∵AA1⊥面ABC,∴AA1⊥AC,AA1⊥AB
又∵AA1=AC,∴四边形AA1C1C是正方形,∴AC1⊥A1C.
∵AB⊥AC,AB⊥AA1,AA1,AC⊂平面AA1C1C,AA1∩AC=A,
∴AB⊥平面AA1C1C.
又∵AC1⊂平面AA1C1C,
∴AB⊥AC1,
∵AB,AC1⊂平面ABC1,AB∩AC1=A
∴A1C⊥平面ABC1.---(5分)
(2)【解析】
∵AA1∥平面BB1C1C,∴点P到平面BB1C1C的距离等于点A到平面BB1C1C的距离
∴,----(9分)
V'=-t(t-1),令V'=0,得t=0(舍去)或t=1,列表,得
t (0,1) 1
V' + -
V 递增 极大值 递减
∴当t=1时,.---(12分)