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Problem 74

Ferris wheel. The model for the height $h$ of a Ferris wheel car is $h=51+50 \sin 8 \pi t$

where $t$ is measured in minutes. (The Ferris wheel has a radius of 50 feet.) This model yields a height of 51 feet when $t=0$ . Alter the model so that the height of the car is 1 foot when $t=0$ .

Answer

$$\csc \left(\frac{2 \pi}{9}\right)=1.5557$$

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## Discussion

## Video Transcript

hides us with a function that is used to model the height of a Ferris wheel. That function is H equals 51 plus 50 sign of eight pi ti where t is expressed in minutes. We're asked to alter this model so that the height of the car is only one foot when t equals zero. So what is the height of the car? Currently, when t equals zero? Well, h of zero equals 51 plus 50. Sign of a pi times zero. Now sign of a pi. Times zero is the same assigned zero, which equals zero. So h of zero equals 51. And that's 50 more feet than we want our function to be at when t equals zero. So what we can do is subtract 50 from our function. I'm gonna call our new model G of tea, and we'll have GFT equals that 51 minus 50 which is just one was 50. Sign a pie teen and we can prove that this is an appropriate model. By plucking in zero for tea, we see that that equals G of zero of one plus 50 times sign of zero that goes straight to zero and G F zero equals one So g of tea. Is there new model where, instead of adding 51 we just add one, and that gives us the appropriate one foot value in T equals zero.

## Recommended Questions

FERRIS WHEEL A Ferris wheel is built such that the height $h$ (in feet) above ground of a seat on the wheel at time $t$ (in seconds) can be modeled by

$h(t) =\ 53 + 50\ sin(\dfrac{\pi}{10}t - \dfrac{\pi}{2})$.

(a) Find the period of the model. What does the period tell you about the ride

(b) Find the amplitude of the model. What does the amplitude tell you about the ride?

(c) Use a graphing utility to graph one cycle of the model.

A Ferris wheel is built such that the height $ h $ (in feet) above ground of a seat on the wheel at time $ t $ (in minutes) can be modeled by

$ h(t) = 53 + 50 \sin \left(\dfrac{\pi}{16} t - \dfrac{\pi}{2}\right) $.

The wheel makes one revolution every $ 32 $ seconds. The ride begins when $ t = 0 $.

(a) During the first $ 32 $ seconds of the ride, when will a person on the Ferris wheel be $ 53 $ feet above ground?

(b) When will a person be at the top of the Ferris wheel for the first time during the ride? If the ride lasts $ 160 $ seconds, how many times will a person be at the top of the ride, and at what times?

You are riding a Ferris wheel. Your height $h$ (in feet) above the ground at any time $t$ (in seconds) can be modeled by $$h=25 \sin \frac{\pi}{15}(t-75)+30$$ The Ferris wheel turns for 135 seconds before it stops to let the first passengers off.

(a) Use a graphing utility to graph the model.

(b) What are the minimum and maximum heights above the ground?

The formula

$$h(t)=125 \sin \left(2 \pi t-\frac{\pi}{2}\right)+125$$

represents the height above the ground at time $t$, in minutes, of a person who is riding a ferris wheel. During the first turn, how much time does a passenger spend at or above a height of 200 feet?

In $1893,$ George Ferris engineered the Ferris Wheel. It was 250 feet in diameter. If the wheel makes 1 revolution every 40 seconds, then the function

$$h(t)=125 \sin \left(0.157 t-\frac{\pi}{2}\right)+125$$

represents the height $h,$ in feet, of a seat on the wheel as a function of time $t,$ where $t$ is measured in seconds. The ride begins when $t=0$

(a) During the first 40 seconds of the ride, at what time $t$ is an individual on the Ferris Wheel exactly 125 feet above the ground?

(b) During the first 80 seconds of the ride, at what time $t$ is an individual on the Ferris Wheel exactly 250 feet above the ground?

(c) During the first 40 seconds of the ride, over what interval of time $t$ is an individual on the Ferris Wheel more than 125 feet above the ground?

Recreation You are riding a Ferris wheel. Your height $h$ (in feet) above the ground at any time $t$ (in seconds) can be modeled by

$h=25 \sin \frac{\pi}{15}(t-75)+30$

The Ferris wheel turns for 135 seconds before it stops to let the first passengers off.

(a) Use a graphing utility to graph the model.

(b) What are the minimum and maximum heights above the ground?

Ferris Wheel A ferris wheel has a radius of $10 \mathrm{m},$ and the bottom of the wheel passes 1 $\mathrm{m}$ above the ground. If the ferris wheel makes one complete revolution every 20 $\mathrm{s}$ , find an equation that gives the height above the ground of a person on the ferris wheel as a function of time.

Ferris Wheel A Ferris wheel has a radius of $10 \mathrm{m},$ and the

bottom of the wheel passes 1 $\mathrm{m}$ above the ground. If the

Ferris wheel makes one complete revolution every $20 \mathrm{s},$ find

an equation that gives the height above the ground of a per-

son on the Ferris wheel as a function of time.

A Ferris wheel is 25 meter in diameter and boarded from a platform that is 1 meter above the ground. The six o'clock position on the Ferris wheel is level with the loading plafform. The wheel completes 1 full revolution in 10 minutes. The function $h(t)$ gives a person's height in meters above the ground $t$ minutes after the wheel begins to tum.

a. Find the amplitude, midline, and period of $h(t) .$

b. Find a formula for the height function $h(t)$

c. How high off the ground is a person after 5 minutes?

Writing to Learn For the Ferris wheel in Exercise 31 , which equation correctly models the height of a rider who begins the ride at the bottom of the wheel when $t=0 ?$

(a) $h=25 \sin \frac{\pi t}{10}$

(b) $h=25 \sin \frac{\pi t}{10}+8$

(e) $h=25 \sin \frac{\pi t}{10}+33$

(d) $h=25 \sin \left(\frac{\pi t}{10}+\frac{3 \pi}{2}\right)+33$

Explain your thought process, and use of a graphing utility in choosing the correct modeling equation.