At the amusement park, you decide to ride the Ferris wheel, which has a maximum height of 100 meters and a diameter of 50 meters. It takes the wheel seven minutes to make one revolution. Write the sinusoidal function, f(t), that models the height of your chair at any time,
we know a = 50, and the period is 7 minutes
2π/k = 7
k = 2π/7
then let's start with the basic shape of
height = 50sin (2π/7)t , where t is in minutes and height is in metres.
but the maximum height is 100 m , so we have to raise our curve 50 m
height = 50sin (2π/7)t + 50
but we probably want our height to be zero when t = 0, so we need a phase shift
height = 50sin(2π/7)(t+d) + 50
50sin(2π/7)(d) + 50 = 0
sin (2π/7)(d) = -1
we know sin 3π/2 = -1 , so
(2π/7)(d) = 3π/2
d = (3π/2)(7/2π) = 21/4
height = 50 sin (2π/7)(t + 21/4) + 50
check:
if t = 0, height = 0 , check!
if t = 1.75, height = 50
if t = 3.5, height = 100
if t = 5.25 , height = 50
if t = 7 , height = 0, all checks out
f(t) = 50 sin ( (2π/7)(t + 21/4) ) + 50
confirmation:
www.wolframalpha.com/input/?i=plot+y+%3D+50+sin+(+(2%CF%80%2F7)(x+%2B+21%2F4)+)+%2B+50
To model the height of your chair at any time, we can use a sinusoidal function. The general form of a sinusoidal function is f(t) = A * sin(B * (t - C)) + D, where:
- A represents the amplitude, which is half the difference between the maximum and minimum values of the function.
- B represents the frequency, which is the number of cycles completed in a given time interval.
- C represents the horizontal shift, which is the phase shift of the function.
- D represents the vertical shift, which moves the entire function up or down.
In this case, the height of your chair at any time can be modeled using the sinusoidal function f(t) = A * sin(B * t) + D, where:
- The amplitude, A, is half the difference between the maximum and minimum height of the Ferris wheel. In this case, the maximum height is 100 meters and the minimum height is 0 meters, so A = (100 - 0) / 2 = 50 meters.
- The frequency, B, is determined by the time it takes for the wheel to make one complete revolution. In this case, it takes the wheel 7 minutes (or 7 * 60 = 420 seconds) to make one revolution. Since one revolution corresponds to a full cycle of the sinusoidal function, the frequency B = 2π / T, where T is the period. The period, T, is the time it takes for the Ferris wheel to complete one cycle. Therefore, T = 420 seconds. Substituting these values, we get B = 2π / 420 ≈ 0.015 rad/s.
- There is no horizontal shift, so C = 0.
- The vertical shift, D, is the midway point between the maximum and minimum height, which is the average of the two. In this case, D = (100 + 0) / 2 = 50 meters.
Putting it all together, the sinusoidal function that models the height of your chair at any time, f(t), is:
f(t) = 50 * sin(0.013 t) + 50
Note that the frequency in the function has been approximated to 0.015 rad/s as mentioned earlier.