1. A Ferris wheel of diameter 16 m rotates at a rate of 0.4 rad/s and is lifted 2m off the ground. If a platform is made so passengers can board at the centre height of the ferris wheel as it goes up,
a. Determine the amount of time it takes to complete one full rotation. [1A]
b. Determine a cosine function that models the height, h in meters, of the car relative to the ground as a function time, t, in seconds. [3A]
c. How long does it take the passenger to reach 14m above the ground for the first time if they started in the middle? Round answer to the nearest tenth of a meter. Be sure to include a therefore statement. [3A]
For a, I got 15.71 seconds. For b, I got h(t) = -8 cos(0.40t) + 10.
I'm really confused on how to do c since it's specifically states from the middle. This was my initial answer:
Since they start in the middle, h = (18 - 14) - 4 = 4
4 = -8 cos(0.40t) + 10
-6 = -8 cos(0.40t)
Let t = cos(0.4t)
-6 = -8 t
-6/-8 = t
t = 0.75 seconds
I don't really think this is correct. Could anyone please help me solve c)?
Your function 10-8cos(0.4t) has the correct period, but is a minimum at t=0. Part (c) of this problem indicates you want to be at axle height when t=0.
That would be 10-8sin(0.4t)
But, since cos(x) = sin(π/2 -x) you want
h(t) = 10 - 8cos(π/2 - 0.4t) = 10 - 8cos(0.4 (π/5 - t))
or, since cos(-x) = cos(x), it might look nicer as
h(t) = 10 - 8cos(0.4 (t - π/5))
That also helps to show the shift of 1/4 period.
Hi! I'm a bit confused on this part of your response: "Part (c) of this problem indicates you want to be at axle height when t=0."
Why is that the case? Also, if h(t) = 10 - 8cos(0.4 (t - π/5)) is the equation, how would you use this to solve for part (c)?
To solve part c) and find the time it takes for the passenger to reach 14m above the ground for the first time, we need to set up and solve the equation for the height of the car relative to the ground.
The cosine function you obtained, h(t) = -8 cos(0.40t) + 10, represents the height of the car as a function of time.
To find the time it takes to reach a height of 14m above the ground, we can set h(t) = 14 and solve for t:
14 = -8 cos(0.40t) + 10
First, let's isolate the cosine term:
-4 = -8 cos(0.40t)
Now, divide both sides by -8:
1/2 = cos(0.40t)
To find the angle whose cosine is 1/2, we can use the inverse cosine (also known as arccosine) function:
0.40t = arccos(1/2)
Using a calculator, we find that arccos(1/2) = π/3. Therefore:
0.40t = π/3
Now, solve for t:
t = (π/3) / 0.40
Calculate the value:
t ≈ 2.356
So, the passenger will reach a height of 14m above the ground for the first time after approximately 2.356 seconds when starting in the middle.
Therefore, the statement is: It will take approximately 2.356 seconds for the passenger to reach a height of 14m above the ground for the first time when starting in the middle.