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A wooden roller coaster contains a run in the shape of a sinusoidal​ curve, with a series of hills. The crest of each hill is 108 feet above the ground. If it takes a car 1.9 seconds to go from the top of a hill to the bottom ​(4 feet off the​ ground), find a sinusoidal function of the form y = A sin(ωt - Φ) + B that models the motion of the coaster train during this run starting at the top of a hill.

Sagot :

lublana

Answer:

[tex]y(t)=52sin(3.3t+\pi/2)+56[/tex]

Step-by-step explanation:

We are given that

Height of crest, h=108 feet

Time for one cycle, T=1.9 s

We have to find a sinusoidal function of the form y = A sin(ωt - Φ) + B that models the motion of the coaster train during this run starting at the top of a hill.

We are given

[tex]y=Asin(\omega t-\phi)+B[/tex]

Amplitude,[tex]A=\frac{108-4}{2}=52 feet[/tex]

[tex]B=A+4=52+4=56 feet[/tex]

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

Using the formula

[tex]\omega=\frac{2\pi}{1.9}=3.3/s[/tex]

We assume,

Horizontal shift [tex]\phi=-\frac{\pi}{2}[/tex]

Substitute the values

[tex]y(t)=52sin(3.3t+\pi/2)+56[/tex]

Hence, the  sinusoidal function  that models the motion of the coaster train during this run starting at the top of a hill is given by

[tex]y(t)=52sin(3.3t+\pi/2)+56[/tex]

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