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]
