Sinusoidal Word Problem

Salt · #1 $Old$ November 5th, 2009, 06:48 PM

A Ferris Wheel has a radius of 12 meters, and the bottom of the wheel passes 3 meters above the ground. At the same time the ride officially begins, Jacky is at some unknown distance from the ground. After 16 seconds, Jacky reaches the highest point on the wheel.

How would you find the sinusoidal equations? Can anyone explain to me how to do this problem? Thanks.

Grandad · #2 $Old$ November 6th, 2009, 02:22 AM

Hello Salt

Welcome to Math Help Forum!

Quote:

Originally Posted by Salt $View Post$

A Ferris Wheel has a radius of 12 meters, and the bottom of the wheel passes 3 meters above the ground. At the same time the ride officially begins, Jacky is at some unknown distance from the ground. After 16 seconds, Jacky reaches the highest point on the wheel.

How would you find the sinusoidal equations? Can anyone explain to me how to do this problem? Thanks.

The equation you need will be in the form

$h=a+b\sin(ct+d)$

where

$h$ is the height above ground in metres
$t$ is the time in seconds
$a,b,c,d$ are constants

You won't be able to solve this completely (you haven't been given enough information) but you can get a set of possible solutions. Here's how:

$\sin\theta$ takes values between $-1$ and $+1$ for all values of $\theta$ , so in the equation above, $h$ takes values from $a-b$ to $a+b$ . Therefore:

$a-b = 3$

$a+b= 3+2\times12 = 27$

This gives:

$a=15, b= 12$

When $t = 16, h = 27$ . So plug these values into the equation using the values of $a$ and $b$ that we've just found:

$27 = 15+12\sin(16c+d)$

$\Rightarrow \sin(16c+d) = 1$

$\Rightarrow 16c+d = 360n+90$ , for any integer $n$ (This is using degrees - but you can write $2n\pi+\pi/2$ in radians if you like)

$\Rightarrow d = 360n+90-16c$

So the equation is of the form

$h = 15+12\sin(ct+360n+90-16c)$

The only other option would be to eliminate $c$ , and express this in terms of $n$ and $d$ . Without more information, that's all you can say.

Grandad