4 * 6 * SIN 30 = 12 cm2

The functional relationship between 2, y and t

S△BCD=6, △PQN and △DQN have the same area, and △DQN is similar to △BCD.

Y = 12 * [(4-t)/4] 2, that is, y = 3/4 (4-t) 2.

3, exist, from 3/4 (4-t) 2. = 24 * 1/4, t=4-2. (Note that another radical 2 of 4+2 is greater than 4, so it is omitted. )

4. According to the conditions, the quadrilateral AMQP with AM=QM is rhombic, and t= 1.6 can be calculated from the proportional line segment, where BM= 1.6 and AM=2.4.

So AP=2.4 cm