The third-tallest Ferris Wheel in the world is the London Eye in England. The height (in metres) of a rider on the London Eye after t minutes can be described by the function h(t) = 67sin [12(t + 0.0223)] + 70.
At what time(s) will the rider be at the bottom of the Ferris wheel?
How long does it take for the Ferris wheel to go through one rotation?
h(t)= 67sin(12(t+0.0223))+70
h(0)=67sin(12(0-0.0223))+70
h(0)= 67sin(12(0.0223)+70
h(0)= 67sin(-0.2676)+70
h(0)= -0.3129224346+70
h(0)= 69.98
So the rider be at the bottom of the Ferris wheel at 69.68 seconds?
To find the time(s) when the rider is at the bottom of the Ferris wheel, we need to identify the values of t that correspond to the lowest points of the sine function within a given interval.
First, let's observe the function h(t) = 67sin [12(t + 0.0223)] + 70. It is a sinusoidal function with a period of 2π/12 or π/6 (since the coefficient of t is 12). The standard form of a sinusoidal function in the form y = A sin(B(t - C)) + D is helpful in identifying the important components:
A: Amplitude, which is 67 in this case.
B: Period, which is π/6.
C: Horizontal shift or phase shift, which is -0.0223.
D: Vertical shift or mean value, which is 70.
The rider will be at the bottom of the Ferris wheel when the rider's height is at its minimum value. In this case, the minimum value will be D - A. Therefore, the minimum height of the rider is 70 - 67 = 3 metres.
The values of t that correspond to the minimum height occur when the sine function takes on the value of -1. Since the general form of the sine function is -1 at t = (2n - 1)π/2, where n is an integer, we can set up the following equation to solve for t:
-1 = sin [12(t + 0.0223)]
Let's solve this equation:
sin [12(t + 0.0223)] = -1
12(t + 0.0223) = -(π/2) + 2nπ, where n is an integer
t + 0.0223 = -(π/24) + 2n(π/12), where n is an integer
t = -(π/24) + 2n(π/12) - 0.0223
Now that we have the general form for t, we can substitute different values of n to find the specific times when the rider is at the bottom of the Ferris wheel.
For example, if we set n = 0, we get:
t = -(π/24) + 0 - 0.0223
t ≈ -0.0223
Similarly, if we set n = 1, we get:
t = -(π/24) + 2(π/12) - 0.0223
t ≈ 0.0923
So, the rider will be at the bottom of the Ferris wheel at approximately t = -0.0223 and t = 0.0923 minutes.
To determine the time it takes for the Ferris wheel to go through one rotation, we need to find the period of the function h(t). We already established that the period is π/6 for the sine function with coefficient 12.
Therefore, it takes π/6 minutes or approximately 0.524 minutes for the Ferris wheel to go through one rotation.