Matthew P. answered • 09/13/19
Computer Science Student Who Loves Teaching Math
Hi there!
To set up this problem, it would be very useful to think about the unit circle, and thinking about sine and cosine as they relate to the x and y positions of the angle measures. Specifically, since we're modeling the height as we go around a circle, we know we're going to want some kind of sine function.
With this in mind, we can start to model our equation. Let's figure out some important numbers first.
- We're given that the diameter is 60 feet, so we know the radius will be 30 feet.
- We're told that the top of the Ferris wheel is 62 feet high, so subtracting the radius we get that the middle of the Ferris wheel is 32 feet high.
- Finally, we're told that the Ferris wheel completes a full rotation (2π radians) in 2 minutes. From this, we can simply write that the Ferris wheel spins at π rad/min.
Now that we have these important numbers, we can start to make our equation! If we consider the unit circle, the height can be expressed as y = r*sinθ, where r is the radius and θ is the angle. The height given by sinθ at the middle of the unit circle is 0; in this problem, the height given would be 32 feet, since that's how tall the middle of the Ferris wheel is. With this, we know that whatever equation we get, we'll have to add 32 to the end of it. We also know that the radius of the Ferris wheel is 30 feet, so our expression for the height "h" will be of the form h(t) = 30sinθ + 32.
With that, all that's left is to figure out how to express the angle θ in terms of t. We figured out earlier that the wheel spins at π rad/min, so for the time t in minutes we can write θ = π * t. We're almost done; we now have h(t) = 30sin(πt) + 32, but we'll see that if we plug in 0 for t, the initial height isn't 62 as we expected. This is because we don't start at θ = 0, but rather θ = π/2. We now have the final piece of our equation: adding π/2 inside the expression for θ.
Our final equation is h(t) = 30sin(πt + π/2) + 32.
I hope this gave you a better understanding of how to model problems like this – let me know if you have any questions!
*Quick footnote, if you're at all interested: in physics, the way we model periodic functions like these that involve a circle is by using the equation x(t) = Acos(ωt + φ), where A is the amplitude, ω is the angular velocity, t is time, and φ is the "phase angle" to satisfy the initial conditions.
Matthew P.
As for the graph, we would just be plotting a sin/cos function starting at the highest point, and then for the values you would just use the rules for period and amplitude, displacement, etc. Let me know if you would like further clarification!09/14/19
Mari M.
wouldnt it be ................. 30 sin pi ( x + pi/2) +3209/15/19
Michael D.
09/16/19
Mari M.
What if it was recipricated. Instead of finding the height from the center of the ferris wheel, your finding the relative height from the ground? What would the new function be.09/19/19
Matthew P.
The expression could definitely use cos instead of sin, you would just have to change the phase angle to match the initial conditions. In this case, the expression would give the full 62 feet when t = 0 because cos(0) = 1. Therefore, if we use cosine, the expression for θ would just be πt, and the whole equation would be 30cos(πt) + 32. I hope that helped!09/14/19