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
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.
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.
at a rate of .4 rad/sec, time taken for one rotation
= 2π/.4 seconds = appr 15.7 sec
amplitude = 8
for period: 2π/k = 15.70796..... , k = .4
starting equation: height = 8cos (.4t)
as it stands, the minimum is -8, we want it to be +2, let's flip it
so ... height = -8cos(.4t) + 10
check for some critical values:
t = 0, height = 2 , that's good
t = 15.708/4 , height = 10 , quarter way up, that's right
t = 15.708/2 , height = 18 , all the way up, correct
...
so when height = 14
-8cos(.4t) + 10 = 14
-8cos(.4t) = 4
-cos(.4t) = .5
cos (.4t) = -.5
I know cos 2π/3 = -.5
.4t = 2π/3
t = 5.23 seconds
You can check this by entering my equation into a webpage like
www.desmos.com/calculator
and checking the values
Hi, i think you equation may have a problem as since it is a cosine function and not a sine function, the d value can't be zero
a) To determine the amount of time it takes to complete one full rotation, we can use the formula:
Time = (2 * pi) / Angular velocity
Given that the angular velocity is 0.4 rad/s, we can substitute these values into the formula:
Time = (2 * pi) / 0.4
Time = 15.7 seconds
Therefore, it takes approximately 15.7 seconds to complete one full rotation.
b) To determine the cosine function that models the height of the car relative to the ground, we need to find the amplitude, period, and vertical shift.
The amplitude is half the difference between the maximum and minimum values. In this case, half of the diameter is 16 / 2 = 8 meters.
The period is the time it takes to complete one full cycle, which we found to be 15.7 seconds in part a).
The vertical shift is the average height of the car. Since the car is lifted 2 meters off the ground, the vertical shift is 2 meters.
Therefore, the cosine function that models the height, h, of the car relative to the ground as a function of time, t, is:
h(t) = 8 * cos((2 * pi * t) / 15.7) + 2
c) To find out how long it takes for the passenger to reach 14 meters above the ground for the first time, we can substitute the value of h(t) into the equation:
14 = 8 * cos((2 * pi * t) / 15.7) + 2
Rearranging the equation, we get:
8 * cos((2 * pi * t) / 15.7) = 12
Dividing both sides by 8, we have:
cos((2 * pi * t) / 15.7) = 12 / 8
cos((2 * pi * t) / 15.7) = 1.5
Since the cosine function reaches its maximum value of 1 at 0 radians, we can ignore that solution. Therefore, we need to find the first positive solution.
Using the inverse cosine, we get:
(2 * pi * t) / 15.7 = invcos(1.5)
Taking the inverse cosine of 1.5 is not possible since it is outside the range of -1 to 1. Hence, there are no solutions for t that would make the height 14 meters above the ground for the first time.
a) To determine the time it takes to complete one full rotation, we need to calculate the period of the rotation. The period of a rotating object is the time it takes for the object to complete one full cycle or revolution. In the case of the Ferris wheel, one full rotation occurs when it completes a 2π radian cycle.
Given that the Ferris wheel rotates at a rate of 0.4 rad/s, we can use this angular velocity to determine the period. The period, T, is defined as the time it takes to complete one full cycle, and it is related to the angular velocity, ω, by the equation:
T = 2π / ω
Plugging in the given value for angular velocity:
T = 2π / 0.4
T = 5π seconds
Therefore, it takes approximately 5π seconds for the Ferris wheel to complete one full rotation.
b) To determine a cosine function that models the height of the car relative to the ground as a function of time, we need to consider the properties of a cosine function and how they apply to this situation.
Since the Ferris wheel is lifted 2m off the ground and its diameter is 16m, the radius of the Ferris wheel is half of its diameter, which is 8m. The highest point the car reaches is 8m above its starting position, and the lowest point is 8m below its starting position.
A cosine function has the general form:
h(t) = A * cos(Bt + C) + D
Where:
- A represents the amplitude of the function (the maximum deviation from the mean height),
- B represents the frequency of the function (related to the angular velocity),
- C represents the phase shift (related to the starting position of the car),
- D represents the vertical shift (related to the initial height of the car).
In this case, since the maximum deviation from the mean height is 8m, the amplitude (A) is 8. The frequency (B) is related to the angular velocity, and since the angular velocity is 0.4 rad/s, the frequency is B = 0.4.
The phase shift (C) represents the starting position of the car. Since the passengers are boarding at the center height as it goes up, the starting position is in the middle of the cycle. This means there is no phase shift, so C = 0.
The vertical shift (D) represents the initial height of the car, which is 2m above the ground. Therefore, D = 2.
Putting it all together, we can write the cosine function that models the height of the car relative to the ground as a function of time:
h(t) = 8 * cos(0.4t) + 2
c) To determine how long it takes for the passenger to reach 14m above the ground for the first time if they started in the middle, we need to find the value of time (t) when h(t) = 14.
Using the cosine function we derived in part b:
14 = 8 * cos(0.4t) + 2
Subtracting 2 from both sides:
12 = 8 * cos(0.4t)
Divide both sides by 8:
1.5 = cos(0.4t)
To find the value of t, we need to take the inverse cosine (also known as the arccosine) of both sides. This can be done using a calculator:
0.4t = arccos(1.5)
Solve for t by dividing both sides by 0.4:
t ≈ arccos(1.5) / 0.4
Using a calculator, t ≈ 5.76 seconds.
Therefore, it takes approximately 5.76 seconds for the passenger to reach 14m above the ground for the first time if they started in the middle.