Question

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.

Answer 1

Answer:

$y(t)=52sin(3.3t+\pi/2)+56$

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

$y=Asin(\omega t-\phi)+B$

Amplitude,$A=\frac{108-4}{2}=52 feet$

$B=A+4=52+4=56 feet$

$\omega=\frac{2\pi}{T}$

Using the formula

$\omega=\frac{2\pi}{1.9}=3.3/s$

We assume,

Horizontal shift $\phi=-\frac{\pi}{2}$

Substitute the values

$y(t)=52sin(3.3t+\pi/2)+56$

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

$y(t)=52sin(3.3t+\pi/2)+56$