Question

A Ferris wheel is 25 meters 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 platform. 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 turn.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?

Answer 1

Answer:

$h(t) = -12.5cos(0.2\pi t)+ 13.5$

$h(5) = 26m$ meters above ground after 5 mins

Step-by-step explanation:

This is a fun question, it requires a bit of visualization and some familiarity with how circles relate to trigonometric functions.

Thinking process:

• Ferris wheel is a circle
• It's starts from the 6 o clock position (the ground)
• the plot of h(t) will start from the 6 o clock position as well

This much information is enough for you to visualize that the plot for h(t) is a trigonometric function (since we're dealing with a circle).

And as the function is NOT starting from the center (i.e 3 o clock position) it is not a sine function.

so a $cos$ function might do the trick! (think of $y=cosx$ ) but remember that the ferris wheel will start from the ground, so $y=-cosx$ ) is a better choice.

• The diameter of the ferris wheel = 25m
• Since its takes 10mins to do a complete revolution the Period = 10 minutes
• rpm (revolution per minute) = $\dfrac{1rev}{10rev} == 0.1\dfrac{rev}{min}$
• there's 1 meter gap between the ground and the ferris wheel.

Solution:

We can start with the plot of just a general cosine function but with the negative sign due to the reasons given above.

$h = -Acos(Bt) + C$

$h$ is the height and since it is a function in terms of $t$. we can rewrite the function as

$h(t) = -Acos(Bt) + C$

here,

$A$ = The Amplitude (distance from the midline to the peak, think of radius of a circle rather than the diameter)

$B$ = Frequency (in $\frac{radians}{unit\,of\,time}$ )

$C$ = Distance of the midline from the x-axis

To covert these into values that are meaningful in our case

$A$ = The radius of the ferris wheel (12.5 m)

$B$ = rpm of the Ferris wheel converted into radians per second. i.e

$\dfrac{0.1rev}{min} * \dfrac{2\pi radians}{1 rev}\\ 0.2\pi \dfrac{rad}{min}$

$B$ = $0.2\pi \dfrac{rad}{min}$

$C$ = since the ferris wheel is above ground at all times, the midline should be atleast be 12.5 meters away from the ground (i.e. radius) but since it is also specified that there's a 1 meter gap between the ground and the ferris wheel C should be:

$C$ = 12.5 + 1  =13.5

Finally, just put all the values in the equation:

$h(t) = -Acos(Bt) + C$

$h(t) = -12.5cos(0.2\pi t)+ 13.5$

FInally, to find h(5)

$h(t) = -12.5cos(0.2\pi (5))+ 13.5$

$h(t) = -12.5(-1)+ 13.5$

$h(t) = 26$ meters above ground