In a small auditorium, the ceiling is designed to reflect sound from the stage to the back balcony. A performer on the stage floor stands at point P directly under a point C on the ceiling that is 5.0m above the floor. A listener sits at point L, 12.0m horizontally away from P on the floor and 4.0m above the floor on the balcony. Near C the ceiling is plane and makes an angle of 30∘ with the horizontal, sloping upward toward the balcony.
(i) Draw a sectional diagram showing the incident and reflected sound rays at C, and mark the angles of incidence and reflection.
(ii) Using the given dimensions, calculate the angle of incidence of the sound at point C.
(iii) Determine the length of the reflected path PC+CL.
(iv) The direct straight-line distance between P and L is 12.7m. Calculate the time difference between the direct sound and the reflected sound reaching the listener. Take the speed of sound as 340m/s.
(v) State whether the listener will hear a distinct echo or only a prolonged sound. Justify your answer with a numerical check against the echo condition.