# Optimizing Ferris Wheel Height: Solving with Transformed Sine Function

• Veronica_Oles
In summary, the formula for modeling the rider's height above the ground as a function of the angle of rotation is y = 9sin(x-90) + 11. The value of a is 9 because the wheel has a diameter of 18m and the value of c is 11 because the rider enters the car at a height of 2m above the ground. The 90 degrees in the formula ensures that the wheel is at its lowest point when the rider enters the car.
Veronica_Oles

## Homework Statement

The Ferris wheel at a carnival has a diameter of 18m and descends to 2m above the
ground at its lowest point. Assume that a rider enters a car at this point and rides the wheel
for two revolutions

(A) model the riders height above the ground vs the angle of rotation using a transformed sime function.

## The Attempt at a Solution

I know the answer is y= 9 sin (x-90) + 11

I know the a is 9 because the 18/2 and the c value is 11 because 2 + 9 = 11 but I don't know how they got to (x - 90)

You need the wheel to be at its lowest point when the person gets in, which is when ##x=0##. So you need the formula to be at a minimum when ##x=0##. What is the minimum value of the sine function, and at what angles does it occur?

[Note: The 90 must be 90 degrees for the answer to be correct]

## What is a transformed sine function?

A transformed sine function is a mathematical equation that describes the relationship between the input variables (in this case, the height of the Ferris wheel) and the output variables (such as the speed or position of the Ferris wheel). It is a type of sinusoidal function that has been altered or transformed in some way, usually through modifications to its amplitude, period, or phase. In the context of optimizing Ferris wheel height, a transformed sine function can help determine the most efficient and safe height for the ride.

## Why is optimizing Ferris wheel height important?

Optimizing Ferris wheel height is important for several reasons. First, it can help maximize the ride experience for passengers by finding the optimal height that provides a balance of speed and smoothness. Secondly, it can help minimize the energy consumption and maintenance costs of the ride. Lastly, optimizing the height can ensure the safety of riders by avoiding excessive forces and stress on the structure of the Ferris wheel.

## How is a transformed sine function used to optimize Ferris wheel height?

A transformed sine function can be used to model the relationship between the height of the Ferris wheel and other variables, such as the speed or position. By analyzing the function and its parameters, such as the amplitude and period, mathematicians and engineers can determine the optimal height that will provide the best ride experience while minimizing energy consumption and stress on the structure of the ride.

## What are some challenges in optimizing Ferris wheel height using a transformed sine function?

One challenge in optimizing Ferris wheel height using a transformed sine function is determining the appropriate parameters for the function. The amplitude, period, and phase of the function can greatly affect the ride experience, so finding the right balance is crucial. Additionally, external factors such as wind and weight distribution of riders can also impact the optimal height. Therefore, real-world testing and adjustments may be necessary to fine-tune the transformed sine function.

## What are the benefits of using a transformed sine function to optimize Ferris wheel height?

Using a transformed sine function to optimize Ferris wheel height offers several benefits. First, it allows for a more scientific and data-driven approach to determining the optimal height, rather than relying on trial and error. Secondly, it can help minimize energy consumption and maintenance costs, making the ride more cost-effective. Lastly, it can ensure the safety and comfort of riders by finding the most efficient and smooth ride experience.

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