why is my answer wrong?

Question

A ferris wheel is 20 meters in diameter and boarded from a platform that is 1 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 4 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).

I thought the equation would be 10 sin (2pie/4(t+pie/2))+11

Answer 1

Hi Samantha, you can also start with cosine as the base function.

At t=0, cos(0) = 1 and is maximum. So, if we use -cos(0) = -1, we start at a minimum just like we're boarding the ferris wheel at its minimum height.

So, minimum height = 1m. Maximum height = 1m + diameter (20m) = 21m. So, the center of the cosine function will be the average of those two (21 + 1)/2 = 11m.

And, the amplitude is then 10m since we go 11 ± 10m.

So, your function will look like h(t) = 11 - 10cos(kt).

Now, what is k? The period of cosine = 2π. The period of the ferris wheel is 4 min.

Set kt = 2π. 4k = 2π, so k = 2π/4 or π/2.

Your final equation is h(t) = 11 - 10cos(πt/2).

Note that this is equivalent to the other experts' answers because of the rule for sin(x - y):

sin(x - y) = sin(x)cos(y) - cos(x)sin(y).

The other experts have sin((2π/4)*(t-1)). This simplifies to sin((πt/2) - (π/2)).

Using the above formula gives: sin(πt/2)cos(π/2) - cos(πt/2)sin(π/2). But, cos(π/2)=0 and sin(π/2)=1.

So, you get 0 - cos(πt/2) = -cos(πt/2).

Answer 2

The six-o'clock position is at t = 0.

Your equation is correct except for the inclusion of the phase shift.

Answer 3

The 6 o'clock position is 1/4 the way through the entire cycle. 6pm, 9pm, 12 pm, 3 pm. So use 2pi/4 as the horizontal offset (phase shift). And subtract this to move the wave back 1/4 of a full cycle:

f(t)=10*sin((2*pi*t/4) -(2*pi/4)) + 11

Which you can simplify to:

f(t)=10*sin((2*pi/4)*(t-1)) + 11

If you graph the above in Desmos, it will verify that you have the right function.