Chapter 4.8, Problem 32E

Solution

a.

To create a model for h .

The equation for the model is $h=-8\cos \frac{\pi t}{10}+9$ .

Given:

Given, Jacob and Emily ride a Ferris wheel at a carnival in Billings, Montana. The wheel has a 16-m diameter and turns at 3rpm with its lowest point 1 m above the ground. Assume that Jacob and Emily’s height h above the ground is a sinusoidal function of time t (in seconds), where $t=0$ represents the lowest point of the wheel.

Calculation:

Since at $t=0$ , the Ferris wheel is at the lowest point, the model is of the form $h=-a\cos \omega t+k$

The diameter of the wheel is 16 m. So, the height of the rider changes from 8 feet below to 8 feet above the center i.e., -8 to 8. Hence the amplitude of the motion is 8.

Hence,

  $a=8\text{ m}$

k is the vertical shift of the motion above the ground.

Here the shift of center of wheel above the ground is 1+8=9 m.

Hence,

  $k=9\text{ m}$

Given the wheel completes 3 rotations in 1 minute i.e., 60 seconds. Hence, each rotation takes 60/3=20 seconds. So, the period is 20 seconds.

The frequency is given by:

  $\begin{array}{c}f=\frac{1}{p}\\ \frac{\omega }{2\pi }=\frac{1}{20}\\ \omega =\frac{2\pi }{20}\\ \omega =\frac{\pi }{10}\end{array}$

Hence, the equation for the model is $h=-8\cos \frac{\pi t}{10}+9$ .

Conclusion:

The equation for the model is $h=-8\cos \frac{\pi t}{10}+9$ .

b.

To draw a graph of h for $0\le t\le 30$ .

The graph of h for $0\le t\le 30$ is:

  

Given:

Given, Jacob and Emily ride a Ferris wheel at a carnival in Billings, Montana. The wheel has a 16-m diameter and turns at 3rpm with its lowest point 1 m above the ground. Assume that Jacob and Emily’s height h above the ground is a sinusoidal function of time t (in seconds), where $t=0$ represents the lowest point of the wheel.

Calculation:

Graphing the equation using a graphing utility for $0\le t\le 30$:

  

Conclusion:

The graph of h for $0\le t\le 30$ is:

  

c.

To estimate the height of Jacob and Emily above the ground at $t=4$ and $t=10$

The height of Jacob and Emily above the ground at $t=4$ and $t=10$ are 6.528 m and 17 m respectively.

Given:

Given, Jacob and Emily ride a Ferris wheel at a carnival in Billings, Montana. The wheel has a 16-m diameter and turns at 3rpm with its lowest point 1 m above the ground. Assume that Jacob and Emily’s height h above the ground is a sinusoidal function of time t (in seconds), where $t=0$ represents the lowest point of the wheel.

Calculation:

The equation of the motion is $h=-8\cos \frac{\pi t}{10}+9$

When $t=4$ :

  $\begin{array}{c}h\left(4\right)=-8\cos \frac{\pi \left(4\right)}{10}+9\\ h\left(4\right)=-8\cos \frac{2\pi }{5}+9\\ h\left(4\right)\approx -8\left(0.309\right)+9\\ h\left(4\right)\approx -2.472+9\\ h\left(4\right)\approx 6.528\end{array}$

When $t=10$ :

  $\begin{array}{c}h\left(10\right)=-8\cos \frac{\pi \left(10\right)}{10}+9\\ h\left(10\right)=-8\cos \pi +9\\ h\left(10\right)=-8\left(-1\right)+9\\ h\left(10\right)=8+9\\ h\left(10\right)=17\end{array}$

Hence, the height of Jacob and Emily above the ground at $t=4$ and $t=10$ are 6.528 m and 17 m respectively.

Conclusion:

The height of Jacob and Emily above the ground at $t=4$ and $t=10$ are 6.528 m and 17 m respectively.