A point on a waterwheel

Name: ______________________ Date: ____________ A point on a waterwheel (6 points)

Graph

Height of a point on a waterwheel

A waterwheel has a radius of 3 meters. Its center is 1 meter above the surface of the water, so the lowest part of the wheel is under water. The wheel turns at a constant speed and completes one full turn every 12 seconds. A point

P

on the rim of the wheel is at the top of the wheel at time

t = 0

seconds. The sinusoidal function

h

models the height of

P

above the surface of the water, in meters, at time

t

; a negative height means

P

is under water. The graph of

h

for one full turn is shown with its axes unscaled. The points

F

G

J

K

and

M

are on the graph:

F

and

M

are highest points,

J

is a lowest point, and

G

and

K

are at the height of the center of the wheel.

Left: a waterwheel whose center is above the water line, with the lower part of the wheel under water. A point P on the rim is at the top of the wheel. Right: an unscaled graph of h(t), the height of P above the water, for one turn of the wheel. The graph starts at its highest point F, falls through G to its lowest point J, which is below the t-axis, rises through K, and ends at the highest point M. G and K are halfway between the highest and lowest heights. The axes have no scale. — LevlPrep original diagram.

(A) Find the coordinates of the points $F$ , $G$ , $J$ , $K$ and $M$ . [2 points]
(B) The function $h$ is given by $h(t)$ $= a\sin(b(t + c)) + d$ , where $a$ , $b$ , $c$ and $d$ are constants. Find values of $a$ , $b$ , $c$ and $d$ . [2 points]
(C(i)) Find the average rate of change of $h$ on the interval $[3, 6]$ , and explain what its sign means in the context of the problem. [1 point]
(C(ii)) Is the rate of change of $h$ increasing or decreasing on the interval $(6, 9)$ ? Explain your reasoning. [1 point]

The question

Height of a point on a waterwheel