A Ferris wheel has a radius of 35m and starts at 2m above the ground. It rotates once every 53 second.
a) determine the cosine equation of the graph, if the rider gets on at the lowest point.
b) what is the height of the rider at 81 seconds?
since cos(x) starts at a maximum, you must have
-cos(x)
The radius is 35, so
y = -35cos(x)
It starts at y=2, so the axle is at y=37:
y = 37-35cos(x)
The period is 53, so
y = 37-35cos(2pi/53 x)
now just do what you need with that.
Thank you so much! This was so helpful! Really appreciate it
a) To determine the cosine equation of the graph, we can start by finding the amplitude, period, and vertical shift of the cosine function.
The amplitude of the cosine function is the distance from the center to the highest or lowest point of the graph. In this case, the radius of the Ferris wheel is 35m, so the amplitude is 35m.
The period of the cosine function is the time it takes for one complete cycle. In this case, the Ferris wheel takes 53 seconds to complete one rotation, so the period is 53 seconds.
The vertical shift of the cosine function is the distance the graph is shifted up or down. Since the rider gets on at the lowest point, the graph is shifted 2m above the ground.
Putting it all together, the cosine equation can be written as:
y = A cos(B(x - C)) + D
Where:
A = amplitude = 35m
B = 2pi / period = 2pi / 53
C = phase shift (in terms of x) = 0 (since the rider gets on at the lowest point)
D = vertical shift = 2m
So, the cosine equation of the graph is:
y = 35 cos((2pi / 53)x) + 2
b) To find the height of the rider at 81 seconds, we can substitute x = 81 into the equation and solve for y.
y = 35 cos((2pi / 53)(81)) + 2
y = 35 cos(1.209) + 2
y ≈ 35(-0.363) + 2
y ≈ -12.72 + 2
y ≈ -10.72
The height of the rider at 81 seconds is approximately -10.72m.
To determine the cosine equation of the graph, we need to consider the general form of a cosine equation: y = A cos(B(x - C)) + D, where A is the amplitude, B is the frequency, C is the phase shift, and D is the vertical shift.
a) In this case, the rider gets on at the lowest point, which means the lowest point corresponds to the y-intercept of the cosine graph. The lowest point occurs when the cosine function is at its maximum value, which is 1.
First, let's find the amplitude and vertical shift:
Amplitude (A) = 35m (radius of the Ferris wheel)
Vertical shift (D) = 2m (initial height above the ground)
The amplitude is the distance from the highest point to the lowest point of the graph, so in this case, it equals the radius of the Ferris wheel.
Now, let's find the frequency:
Frequency (B) = 2π / Period
Period = 53 seconds
Frequency (B) = 2π / 53
Lastly, let's find the phase shift:
Since the rider gets on at the lowest point, the phase shift (C) is 0.
Putting it all together, the equation for the graph is:
y = 35 cos((2π / 53) x)
b) To find the height of the rider at 81 seconds, we substitute x = 81 into the equation and solve for y:
y = 35 cos((2π / 53) x)
y = 35 cos((2π / 53) * 81)
y ≈ 35 cos(4.868 radians)
Use a scientific calculator to evaluate the cosine of 4.868 radians. The result will give you the height of the rider at 81 seconds.