Chapter 6.5, Problem 43E

(a)

To determine

To find: The lowest height of a seat.

Answer to Problem 43E

21 feet below the center

Explanation of Solution

Given information: As you ride a Ferris wheel, the height that you are above the ground varies periodically. Consider the height of the center of the wheel to be the equilibrium point. Suppose the diameter of a Ferris wheel is 42 feet and travels at a rate of 3 revolutions per minute. At the highest point, a seat on the Ferris wheel is 46 feet above the ground.

Calculation:

If the center of the wheel is the zero point and the diameter is 42 then the radius is 21.

  $\frac{42}{2}=21$

And at the bottom you are 21 feet below the center.

(b)

To determine

To find: The equation of the midline.

Answer to Problem 43E

  $y=25$

Explanation of Solution

Given information: As you ride a Ferris wheel, the height that you are above the ground varies periodically. Consider the height of the center of the wheel to be the equilibrium point. Suppose the diameter of a Ferris wheel is 42 feet and travels at a rate of 3 revolutions per minute. At the highest point, a seat on the Ferris wheel is 46 feet above the ground.

Calculation:

The diameter is 42, so that means the radius is 21.

  $\frac{42}{2}=21$

The maximum height minus the diameter will give you the height off the ground, which is 4.

  $46-42=4$

The midline is going to be the Ferris wheel’s height from the going added to tis radius, the midline is 25 feet.

  $4+21=25$

Thus,

  $y=25$

(c)

To determine

To find: The period of the function.

Answer to Problem 43E

Period is 20 seconds

Explanation of Solution

Given information:As you ride a Ferris wheel, the height that you are above the ground varies periodically. Consider the height of the center of the wheel to be the equilibrium point. Suppose the diameter of a Ferris wheel is 42 feet and travels at a rate of 3 revolutions per minute. At the highest point, a seat on the Ferris wheel is 46 feet above the ground.

Calculation:

If the wheel rotates 3 times a minute then it rotates every 20 seconds.

  $\frac{1}{3}\times 60=20$

So the period is 20 seconds.

(d)

To determine

To write: A sine equation to model the height of a seat that was at the equilibrium point heading upward when the ride began.

Answer to Problem 43E

  $h\left(t\right)=25+21\sin \left(\frac{\pi t}{10}\right)$

Explanation of Solution

Given information: As you ride a Ferris wheel, the height that you are above the ground varies periodically. Consider the height of the center of the wheel to be the equilibrium point. Suppose the diameter of a Ferris wheel is 42 feet and travels at a rate of 3 revolutions per minute. At the highest point, a seat on the Ferris wheel is 46 feet above the ground.

Calculation:

Determine the amplitude

  $\begin{array}{l}\left|A\right|=\frac{42}{2}=21\\ A=\pm 21\end{array}$

Since the seat is heading upward when the ride began, A must be positive, so

A = 21

Determine h and k

  $\frac{2\pi }{k}=20\to k=\frac{2\pi }{20}=\frac{\pi }{10}$

Conclude the sine equation

  $\begin{array}{l}h\left(t\right)=h+A\sin \left(\frac{\pi t}{10}\right)\\ \Rightarrow h\left(t\right)=25+21\sin \left(\frac{\pi t}{10}\right)\end{array}$

(e)

To determine

To find: When the seat will reach the highest point for the first time, according to the model.

Answer to Problem 43E

5 seconds

Explanation of Solution

Given information: Entertainment: As you ride a Ferris wheel, the height that you are above the ground varies periodically. Consider the height of the center of the wheel to be the equilibrium point. Suppose the diameter of a Ferris wheel is 42 feet and travels at a rate of 3 revolutions per minute. At the highest point, a seat on the Ferris wheel is 46 feet above the ground.

Calculation:

If the wheel rotates 3 times in a minute then it rotates every 20 seconds and if we start at the equilibrium point, it will reach the maximum height after a quarter of a revolution,

  $\frac{1}{3}\times \frac{1}{4}=\frac{1}{12}\times 60=5$

Thus, after 5 seconds

(f)

To determine

To find: The height of the seat after 10 seconds, according to the model.

Answer to Problem 43E

0 feet (25 feet above the ground)

Explanation of Solution

Given information: As you ride a Ferris wheel, the height that you are above the ground varies periodically. Consider the height of the center of the wheel to be the equilibrium point. Suppose the diameter of a Ferris wheel is 42 feet and travels at a rate of 3 revolutions per minute. At the highest point, a seat on the Ferris wheel is 46 feet above the ground.

Calculation:

If the wheel rotates 3 times in a minute then it rotates every 20 seconds so the period is 2 seconds.

So after 10 seconds it has done half a revolution which means it will be back at the same height that it started at.

  $\frac{10}{20}=0.5$

Thus, 0 feet (25 feet above ground)