A double Ferris wheel has a 50-foot bar attached to a main support at the
center of the bar. The center is 40 feet off the ground. The bar revolves
counterclockwise once every 60 seconds. Attached to each end of the bar is a
wheel with a radius of 10 feet. If the main support bar were stationary, each
of the two wheels would complete a counterclockwise revolution in 15 seconds.
Assume that at time t = 0 a rider is at a point the farthest to the right as
possible.
5. Write an equation to describe the rider’s height y at time t. Hint: the height
above the center is the sum of the heights of the two pink triangles.
6. Write an equation to describe the rider’s horizontal position x at time t. As-
sume that the x-position of the center of the bar is 0 feet.
7. What is an appropriate interval of t values that will capture an entire ride?
8. Use Desmos grapher, or another graphing app, to draw the path of the rider. On
Desmos, you can do this by entering your expression for x and your expression
for y, in parentheses separated by commas. Forexample, if you have x =52
cos(2π t) + 60 and y = 52 sin( 2π t) + 60 then you can type in (52 cos( 2π t) + 3
60, 52 sin( 2π t) + 60) and enter in appropriate bounds for t to see the shape of 3
the path. Take a screenshot of the path and include it.
9. How do the equations of the rider change if the main support bar is rotating
counterclockwise and the wheels are independently rotating clockwise? Graph
the new path and include a picture. Compare the subjective experience of the rider
on each of the two rides.
2
Write an equation to describe the rider’s height y at time t.
the 50-ft bar is the diameter of the large wheel, so its amplitude is 25
since its period is 60 seconds, our formula for the end of the bar is
The end of the bar starts at y=40, so
let the center of the machine be at (0,40)
y = 40+25sin(π/30 t)
x = 35cos(π/30 t)
Each of the small wheels has amplitude 10 and period 15, so the positions are finally
y = 40+25sin(π/30 t) + 10sin(2π/15 t)
x = 35cos(π/30 t) + 10cos(2π/15 t)
Thank you for the clarification. The equations for the rider's height y and horizontal position x at time t are as follows:
y = 40 + 25sin(π/30 t) + 10sin(2π/15 t)
x = 35cos(π/30 t) + 10cos(2π/15 t)
These equations take into account the movements of the larger Ferris wheel (main support bar) and the smaller wheels attached to each end of the bar. The rider's position is determined by the combination of these movements in both the vertical and horizontal directions.
To capture an entire ride, an appropriate interval of t values would be from 0 to 60 seconds, as this represents one full rotation of the main support bar.
To draw the path of the rider on Desmos or another graphing app, you can enter these equations for x and y, using the given intervals of t values, to see the shape of the path. Once you have plotted the path, take a screenshot to include in your work.
For the second part of the question, when the main support bar is rotating counterclockwise and the wheels independently rotating clockwise, the equations for the rider's height y and horizontal position x would change accordingly. You can use the new equations provided above to graph the new path and compare it to the original path depicted in the screenshot. Consider how the rider's experience would differ on each ride based on the changes in movement and orientation.
To write an equation for the rider's height y at time t, we can consider the two pink triangles formed by the rider and the center of the bar. The height of each triangle can be represented by the radius of the wheel (10 feet) and the horizontal position x of the rider.
Let h be the vertical distance from the ground to the rider's position on the bar at time t. At time t=0, the rider is at the farthest right point on the right wheel, which is 10 feet off the ground. As the bar rotates counterclockwise and the rider moves, the height of the rider can be calculated as follows:
For the pink triangle on the left side (when h is positive):
h = 10sin(2πt/15)
For the pink triangle on the right side (when h is negative):
h = -10sin(2πt/15)
Combining both triangles, the total height y of the rider at time t is:
y = 40 + 10sin(2πt/15) - 10sin(2πt/15)
y = 40
Therefore, the equation to describe the rider's height y at time t is simply y = 40 feet, as the rider's height remains constant at 40 feet above the ground throughout the ride.