Double Ferris WheelOn this web page you can explore the motion of a double Ferris
wheel. This will help you solve Exercise 12 in Lesson 10.5 of Discovering
Advanced Algebra: An Investigative Approach, and it might give you
some ideas for how to create your own simulation using The Geometer's
Sketchpad.
Each wheel has diameter 20 ft and takes 20 s to make a single
rotation. The
center of the lower wheel is 12 ft above the ground, and the center of
the upper wheel is 34 ft above the ground. The center of the twowheel
set is 23 ft above the ground and takes 30 s to rotate once.
Sketch
The sketch below shows a model of the double Ferris wheel. Press the
buttons to rotate just the two wheels, just the twowheel set, or
both rotations. Press a button again to stop the motion. Press Start Over to return Sandra to the
bottom position. (Note:
Depending
on your computer and web browser, the wheels may not rotate at exactly
the correct speeds.)
Investigate
 Sandra gets on at the foot of the bottom wheel. Press Rotate Small Wheels and watch
Sandra's height above the center
of the bottom
wheel as the wheels rotate. Sketch a graph and write an equation that
models her height above the center of this wheel as a function of time.
 The entire ride (the twowheel set) starts revolving at the same
time that the two smaller wheels begin to rotate. Stop the small
wheels, press Rotate TwoWheel Set,
and watch the height of the
center of Sandra's wheel above the ground. Sketch a graph and write an
equation that models the height of the center of Sandra's wheel as a
function of time.
 Stop the twowheel set, press Start
Over to return Sandra
to
the bottom, then press Both Rotations.
Watch Sandra's height above
the ground. Sketch a graph of her height above the ground as a function
of time.
 Because the two motions occur simultaneously, you can add the two
equations (from questions 1 and 2) to write a final equation for
Sandra's position.
Write this equation. Graph the equation on your calculator and compare
it to the sketch you made.
 During a 5 min ride, for how many distinct time periods is Sandra
within 6 ft of the ground? How can you use the sketch to answer this
question? How can you use the graph to answer this question?
Sketch
This sketch allows you to trace Sandra's position above the ground
as a function of time. You can rotate just the small wheels, just the
twowheel set, or both rotations. To start over, click a button again
to stop the motion, press Start Over
to return Sandra to the bottom
position, and then click X to
erase
the traces.
Investigate
 Does each graph agree with your results from the first part of
this exploration? If any of the graphs are different, explain why they
are different.
 After how many seconds does the motion of the entire ride (Both Rotations) repeat itself? How
is this shown on the graph?
 Trace Both Rotations to find how many distinct time
periods
Sandra is within 6 ft of the ground during a 2 min ride. How can you
use recursive thinking to extend your answer to 5 min? 10 min?
 How would Sandra's position above the ground change if her small
wheel is first rotated to let other people on the ride? Start over and
press Rotate Small Wheels to
move Sandra to a different position,
then press Rotate TwoWheel Set
to move the entire ride. How does the
graph compare to the graph of when Sandra started at the bottom?
Discuss
any similarities or differences between the two graphs.
