Respuesta :
The distance from the bottom of the wheel to a point on the ground can be modeled by a sine function. h(t)=a*sin(kt)+b. b is the height of the wheel, or 4m. a is the radius of the wheel, 50/2=25m. 2pi/k is the 5sinof thw revolution, so 2pi/k=6/3=2, k=pi. h(t)=25sin(pi*t)+4.
Answer:
The required function is [tex]y=-25\cos (\pi x)+29[/tex].
Step-by-step explanation:
The general form of a cosine function is
[tex]y=A\cos (Bx+C)+D[/tex] .... (1)
where, A is amplitude, [tex]\frac{2\pi}{B}[/tex] is period, C is phase shift and D is mid line.
It is given that Ferris wheel that sits four meters above the ground and has a diameter of 50 meters. So the minimum value of the function is 4 and maximum value of the function 54 meters.
[tex]D=Midline=\frac{Maximum+Minimum}{2}=\frac{4+54}{2}=29[/tex]
It takes six minutes to do three revolutions on the Ferris wheel.
[tex]Period=\frac{6}{3}[/tex]
[tex]Period=2[/tex]
[tex]\frac{2\pi}{B}=2[/tex]
[tex]B=\pi[/tex]
Phase shift is not given so C=0.
Substitute B=π, C=0 and D=29 in equation (1).
[tex]y=A\cos (\pi x+0)+29[/tex]
[tex]y=A\cos (\pi x)+29[/tex] ... (2)
He enters the ride at the low point when t = 0. It means the function passes through the point (0,4).
[tex]4=A\cos (\pi (0))+29[/tex]
[tex]4=A+29[/tex]
[tex]4-29=A[/tex]
[tex]-25=A[/tex]
The amplitude is -25. Put this value in equation (2).
[tex]y=-25\cos (\pi x)+29[/tex]
Therefore the required function is [tex]y=-25\cos (\pi x)+29[/tex].
