A Ferris wheel completes one revolution every 90 s. The cars reach a
maximum of 55 m above the ground and a minimum of 5 m above the ground.
The height, h, in metres, above the ground can be modeled using a sine function,
where t represents time, in seconds. Assume ride starts when you get on the ride.
a) Draw a sketch showing one rotation of the Ferris wheel.
b) Determine an equation that models the height of a rider.
c) Determine the height of a rider after 70 seconds.
sin(kx) has period 2π/k, so since we have period 90, k = π/45
y = a sin(π/45 t+b)) + c
sin(x) has a minimum at t = 3π/2, and our ferris wheel has a minimum at t=0, so
y = a sin(π/45 t + 3π/2)) + c
The axle is at (55+5)/2 = 30, so
y = a sin(π/45 t + 3π/2) + 30
The amplitude is (55-5)/2 = 25, so
y = 25 sin(π/45 t + 3π/2)) + 30
or
y = 25 sin(π/45 (t + 135/2)) + 30
a) To draw a sketch showing one rotation of the Ferris wheel, we can plot the height of the rider as a function of time. The axis of the graph will represent time (t) in seconds, and the vertical axis will represent the height (h) in meters.
b) The height of the rider can be modeled using a sine function. The general form of a sine function is:
h(t) = A * sin(B * t + C) + D
where:
A = amplitude (half the peak-to-peak height, so A = (55 - 5) / 2 = 50 / 2 = 25)
B = 2π / T, where T is the period (T = 90 s)
C = phase shift (0, in this case)
D = vertical shift (the minimum height, 5)
Therefore, the equation that models the height of a rider is:
h(t) = 25 * sin((2π / 90) * t) + 5
c) To determine the height of a rider after 70 seconds, substitute t = 70 into the equation and solve:
h(70) = 25 * sin((2π / 90) * 70) + 5
Now we can calculate the value of h(70).
a) To draw a sketch, we need to visualize how the height of the rider changes over one complete revolution of the Ferris wheel. We know that the maximum height is 55 m and the minimum height is 5 m.
Let's assume the center of the Ferris wheel is at height 0 m. We can draw a horizontal line to represent the ground. To represent the height, we can draw a sinusoidal wave pattern, starting from the minimum height and reaching the maximum height.
b) The equation that models the height of a rider can be written as:
h(t) = A*sin(B(t - C)) + D
In this equation:
- A represents the amplitude, which is half the difference between the maximum and minimum heights. In this case, A = (55-5)/2 = 25 m.
- B represents the period, which is given by the formula 2π / T, where T is the time taken to complete one revolution. In this case, T = 90 s. So, B = 2π / 90.
- C represents the phase shift, which determines the starting point of the sinusoidal wave. Since we are assuming the ride starts when you get on, the phase shift is 0.
- D represents the vertical shift, which is the average of the maximum and minimum heights. In this case, D = (55 + 5)/2 = 30 m.
Therefore, the equation that models the height of a rider is:
h(t) = 25*sin((2π / 90) * (t - 0)) + 30
c) To determine the height of a rider after 70 seconds, we can substitute t = 70 into the equation:
h(70) = 25*sin((2π / 90) * (70 - 0)) + 30
Using a calculator or software, evaluate the expression:
h(70) = 25*sin((2π / 90) * 70) + 30
The result will give you the height of a rider after 70 seconds.