The Great Laxey Wheel, located on the Isle of Man, is the largest working water wheel in the world. The highest point of a bucket on the wheel is 70.5 feet above the viewing platform, and the lowest point is 2 feet below the viewing platform. The wheel makes a complete turn every 24 seconds. Write a model for the height h (in feet) of the bucket as a function of time t (in seconds) given that the bucket is at its lowest point when t

Answer:

h(t)=-36.25cos(0.2618t)+34.25

Step-by-step explanation:

We can start by drawing a sketch of what the whaterwheel's height over time looks like. (See attached picture.)

We can see on the graph that at t=0, the bucket will be located at its lowest point. So in this case we can make use of a cosine function. Cosine functions look like this:

[tex]y=Acos(\omega t + \phi)+b[/tex]

where:

A=amplitude

[tex]\omega[/tex] = angular speed

[tex]\phi[/tex]=phase shift

b= vertical shift.

So let's find each of the necessary data:

The amplitude is the distance between the highest point of the trajectory and the middle point of the sine wave, so:

[tex]A=\frac{highest-lowest}{2}[/tex]

[tex]A=\frac{70.5ft-(-2ft)}{2}[/tex]

A=36.25 ft

since the trajectory starts at its lowest point when t=0 we will make the amplitude a negative amplitude, so:

A=-36.25ft

Next, we can find the angular speed:

[tex]\omega = \frac{2\pi}{T}[/tex]

Where T is the period (the time it takes the wheel to make one whole turn) so:

[tex]\omega=\frac{2\pi}{24}[/tex]

[tex]\omega=0.2618 rad/s[/tex]

the phase shift in this case is zero since the graph starts at the lowest point.

the vertical shift is the distance between the x-axis and the middle point of the graph. So we find the midpoint between the lowest and the highest point of the graph:

[tex]b=\frac{highest+lowest}{2}[/tex]

[tex]b=\frac{70.5ft-2ft}{2}[/tex]

b=34.25ft

so we can now input all this data into the formula to get:

[tex]h(t)=-36.25cos(0.2618 t)+34.25[/tex]

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