This question is really difficult and troublesome, and I'm getting impatient. This topic examines the properties of rectangles and squares, similar triangles's judgment and properties, the definition of acute trigonometric function, the midline theorem of triangles, Pythagorean theorem and other knowledge, examines the area of pentagons by cutting complement method, examines the range of values obtained by critical value method, and examines the mathematical ideas of classification discussion. It's really no ordinary trouble. Look at the answer Kiss and hug

In the rectangle ABCD, AB=4, BC=3, point P starts from point A and moves along the dotted line AD-DO-OC to the end point C at the speed of/kloc-0 per unit length per second. When point P does not coincide with point A, the intersection point P is PQ perpendicular to point Q, with PQ as the edge, square PQMN is made to the right, and the overlapping area of square PQMN and triangle ABD is S.

(2) Write directly the value range of t when the point O is in the square PQMN;

(3) When the point P moves on the broken line AD-DO, find the functional relationship between S and T;

(4) Write the value of t directly when the straight line DN bisects the area of the triangle BCD.