Question

A ferris wheel is 35 meters in diameter and boarded at ground level. The wheel makes one full rotation every 8 minutes, and at time t = 0 you are at the 9 o'clock position and descending. Let f(t) denote your height (in meters) above ground at t minutes. Find a formula for f(t).

Answer 1

f(t) = -17.5sin((t/4)*pi) + 17.5
Because the ferris wheel is a circle you can use sine and cosine. As we are descending and in the middle of the descent the use of negative sine makes the most sense.
Period = 2pi.
Our period is 8 minutes.
8 minutes*w = 2pi
w = (t/4) * pi.
The 17.5 at the end is to take into account that we are 17.5 meteres up in the air at the start.

Answer 2

To model the function of the Ferris wheel, we make use of a sine model, because, the 2D shape of the Ferris wheel is a circle.

The function is: $\mathbf{f(t) = 17.5sin(\frac{\pi}{4}(t - 6)) + 17.5}$ or $\mathbf{f(t) = 17.5sin(\frac{\pi}{4}(t - 2)) + 17.5}$

The diameter is given as:

$\mathbf{d = 35}$

So, the amplitude (A) of the function is:

$\mathbf{A = \frac{d}{2}}$

$\mathbf{A = \frac{35}{2}}$

$\mathbf{A = 17.5}$

And the vertical shift is:

$\mathbf{D = 17.5}$

A sine function is represented as:

$\mathbf{f(x) = Asin(B(x - C)) + D}$

So, we have:

$\mathbf{f(x) = 17.5sin(B(x - C)) + 17.5}$

A complete rotation of a circle is 360 degrees.

So, the period (T) is:

$\mathbf{T = 360^o}$

Express as radians

$\mathbf{T = 2\pi}$

It rotates every 8 minutes.

So, the number of rotation in 1 period is:

$\mathbf{8B = 2\pi}$

Divide both sides by 8

$\mathbf{B = \frac{\pi}{4}}$

So, we have:

$\mathbf{f(x) = 17.5sin(\frac{\pi}{4}(x - C)) + 17.5}$

At 9'O clock, the position of the ferry will be 6/8, and at a height of 17.5

So, we have:

$\mathbf{17.5 = 17.5sin(\frac{\pi}{4}(6 - C)) + 17.5}$

Subtract 17.5 from both sides

$\mathbf{0 = 17.5sin(\frac{\pi}{4}(6 - C)) }$

Divide both sides by 17.5

$\mathbf{0 = sin(\frac{\pi}{4}(6 - C) )}$

Take arc sin of both sides

$\mathbf{\frac{\pi}{4}(6 - C) = 0}$ or $\mathbf{\frac{\pi}{4}(6 - C) = \pi}$

Divide both sides

$\mathbf{6 - C = 0}$ or $\mathbf{6 - C = 4}$

Solve for c

$\mathbf{C = 6}$ or $\mathbf{C = 2}$

So, the function is:

$\mathbf{f(x) = 17.5sin(\frac{\pi}{4}(x - 6)) + 17.5}$ or $\mathbf{f(x) = 17.5sin(\frac{\pi}{4}(x - 2)) + 17.5}$

Substitute t for x

$\mathbf{f(t) = 17.5sin(\frac{\pi}{4}(t - 6)) + 17.5}$ or $\mathbf{f(t) = 17.5sin(\frac{\pi}{4}(t - 2)) + 17.5}$

Read more about sine functions at:

https://brainly.com/question/1368748