Suppose you wanted to model the height above the ground of the tip of the blade of a windmill over time in seconds using a sinusoidal function. The windmill takes 45 seconds to complete one revolution and the tip of
the blade is initially at a height of 1.5m above the ground. Provide an equation of such a sine function that will ensure that the minimum height of the tip of the blade from the ground is 1.5 m. Note that the maximum
height can be any reasonable value of your choice. [4C]
Explain why your equation works.
To model the height of the tip of the windmill blade above the ground over time using a sinusoidal function, we can start by understanding the characteristics of the given scenario.
1. The windmill takes 45 seconds to complete one revolution. This means that the blade will go through one complete cycle every 45 seconds.
2. The tip of the blade is initially at a height of 1.5 meters above the ground. This gives us the starting point for our function.
3. We want to ensure that the minimum height of the tip of the blade from the ground is 1.5 meters.
To create a sinusoidal function, we can rely on the general form:
f(t) = A*sin(B(t-C)) + D
where:
- A represents the amplitude, which determines the range of the function (from the minimum to the maximum height).
- B affects the period of the function, determining how quickly it oscillates.
- C shifts the function horizontally, affecting the starting point.
- D represents the vertical shift of the function.
Since the blade takes 45 seconds to complete one revolution, the function should have a period of 45 seconds. This means that B = (2π)/45.
To ensure the minimum height is 1.5 meters, we need to set A = maximum height - 1.5. The maximum height value can be chosen based on the desired range.
With these considerations, the equation for the desired sinusoidal function becomes:
f(t) = (maximum height - 1.5)*sin((2π/45)(t - C)) + 1.5
By choosing appropriate values for the maximum height and C, we can set the vertical shift and the range of the function to fulfill the requirements of the problem.