Given:
The radius of the Ferris wheel is 9.5 m.
The wheel rotates fully once every 10 seconds.
Aim:
We need to find the equation that represents the height of a rider with respect to time.
Explanation:
The form of sinusoidal eqaution is
[tex]y=A\text{ }\cos \text{(}B(x-C))D[/tex][tex]A=\frac{\text{maximum}-\text{minimum}}{2}[/tex]
The height from the ground to the bottom of the Ferris wheel is zero.
The maximum height is 9.5+9.5 =19 m.
The minimum is 0 m.
Substitute maximum =19 and minimum =0 in equation A.
[tex]A=\frac{19-0}{2}=9.5[/tex][tex]B=\frac{2\pi}{P\text{eriod}}[/tex]
Substitute period = 10 seconds in the equation.
[tex]B=\frac{2\pi}{10}=\frac{\pi}{5}[/tex][tex]D=\frac{Maximum+Minimum}{2}[/tex]
Substitute maximum =19 and minimum =0 in equation D.
[tex]D=\frac{19+0_{}}{2}=9.5m[/tex]
Substitute know values in the equation, we get
[tex]y=0.5\cos \text{(}\frac{\pi}{5}(x-C))+9.5[/tex]
Set C=0 in the equation, we get
[tex]y=9.5\cos \text{(}\frac{\pi}{5}x)+_{}9.5[/tex]
The graph of the eqaution is
The value of A should be negative.
The equation is of the form
[tex]y=-9.5\cos \text{(}\frac{\pi}{5}x)+_{}9.5[/tex]
where y represents the height from the bottom of the wheel to the rider and x represents the time taken.
The x-intercepta are (10,0), (20,0) ,(30, 0 ) and so on.
The y-intecept is (0,0).
The amplitude is the exact value of A.
[tex]|A|=|9.5|=9.5[/tex]
The period of the function is
[tex]B=\frac{\pi}{5}[/tex]
The amplitude of the given equation is 9.5.
