The figure shows a pencil sharpener on a level surface. Point B is located at the end of a
handle that is 2 inches from the center of rotation. The handle rotates in a clockwise
direction and completes 2 rotations every second. At time t = 0 seconds, point B is
located directly below the center of rotation. The center of rotation is 3 inches above the
level surface on which the pencil sharpener sits. As the handle rotates, the distance
between point B and the level surface periodically increases and decreases. The sinusoidal
function h models the distance, in inches, between point B and the level surface as a function of time t in seconds.
(A) The graph of h and its dashed midline for two full cycles is
shown. Five points, F, G, J, K, and P are labeled on the graph.
No scale is indicated, and no axes are presented. Determine
possible coordinates (t, h(t)) for the five points:
F, G, J, K, and P
(B) The function h can be written in the form h(t) = a sin(b(t + c)) + d. Find values of the constants a, b, c, and d.
(C) Refer to the graph of h in part (A). The t-coordinate of J is t1, and the t-coordinate of K is t2.
(i) On the interval (t1, t2), which of the following is true about h?
a. h is positive and increasing. b. h is positive and decreasing.
c. h is negative and increasing. d. h is negative and decreasing.
(ii) Describe how the rate of change of h is changing over the interval (t1, t2).
