Precalc Problem

Question

 ferris wheel is 20 meters in diameter and boarded from a platform that is 2 meters above the ground. The six o'clock position on the ferris wheel is level with the loading platform. The wheel completes 1 full revolution in 10 minutes. The function h = f(t) gives your height in meters above the ground t minutes after the wheel begins to turn. Write an equation for h = f(t).

Answer 1

Model f(t) = Asin(B(t+C))+D

D = platform height + radius = 12

B=2π/period = 2π/10 = π/5

A=radius=10

To Solve for C:

f(0) = 2 = 10sin((π/5)(0+C))+12

sin(πC/5)=-10/10=-1

sin^-1(-1)=3π/2

πC/5 = 3π/2 --> C=15/2

f(t) = 10sin((π/5)(t+15/2)+12

f(t) = 10sin(πt/5 +3π/2)+12

Checking

f(0) = 2 --> Platform height

f(5) = 10sin(π/2)+12=22 --> top of the Ferris wheel

f(5/2)=10sin(2π)+12 = 0+12=12 --> At 3:00 position

Answer 2

Hi Everett,

I don't know you, so I have to guess what part(s) of this question have got you stumped.

First, the 6 o'clock position refers to the lowest part of the wheel.

Second, as the wheel goes around and around the heights of the riders repeat in a periodic way and a graph will look like a sine or cosine wave, yes?

Since the riders board the Ferris wheel at the bottom, when t = 0, a cosine function would be better than a sine function, because they start at 0. In fact, since this is the LOWEST position, an inverted cosine function would perfect. Therefore, the amplitude coefficient will be negative.

Do you agree that the amplitude of the function would be 10, the radius of the Ferris wheel?

Do you see that the center of the Ferris wheel is 12 meters off the ground?

And that this height represents a vertical shift.

So far, we can see that h(t) = -10 Cos (Bt) +12, yes.

But how to we determine B?

Your teacher may have given you a formula to memorize, but I'm going to show a way to proceed even if you can't remember the formula.

We know that one complete rotation occurs when Bt = 2π.

We also know that this occurs when t = 10.

Substitute 10 for t and solve for B.

B = (2π)/10 and this reduces to π/5.

Our final function is h(t) = -10 Cos (πt/5) + 12.

I hope this helps.