A clock with an hour hand that is 15 inches long is hanging on a wall. At noon, the distance between the tip of the hour hand and the ceiling

is 23 inches. At 3 P.M., the distance is 38inches; at 6 P.M., 53 inches; at 9 P.M., 38 inches; and at midnight the distance is again 23 inches.Let D represent the distance between the tip of the hour hand and the ceiling t hours after noon.An equation that models the distance between the tip of the hour hand and the ceiling as a function of time t is D(t)=d+asin(k(t –b). The distance D will vary with time. Using the axes provided below, sketch a graph to show how the distance between the tip of the hour hand and the ceiling for a 24 hour time period.а. b. Use the graph in part a to find the center line (sinusoidal axis) of the graph. c. What is the vertical shift d? Explain why. d.Find the amplitude of this function. Explain how you found it. What is the period of this function? Explain why. f. What is the exact value of k ? Justify your answer algebraically. g. Find the horizontal shift (phase shift) of this function. Explain why. h. Write the equation that describes this motion using the model D(t)=d+asin(k(t -b).(6 points) i. What is the distance D between the ceiling and the tip of the hour hand at 5:00 P.M? Show your work and give your answer in exact and approximating form.(5 points) j. Use the equation D=d+asin(k(t –b)) to write t as a function in terms of D. Justify your answer algebraically.(7 points) 1 According to this model, when would be the first time that the distance between the tip of the hour hand and the ceiling is 48 inches? Justify your answer algebraically and round your answer to two decimal places.(6 points)