A ferris wheel has a radius of 13 m. It rotates once every 24 seconds. A passenger gets on at the bottom
of the wheel from a ramp which is one metre above ground level.
a) If the height of the passenger is measured from the ground, determine an equation for the height of the
passenger as a function of time in the form of h(t)= acos(bt)+d
b) To the nearest metre, find the height of the passenger after 55 seconds.
since cos(t) has a max at t=0, and we want the passenger to start at the bottom, a will be negative.
r=13, so
h(t) = -13cos(bt)+d
h(0) = 1, so
h(t) = -13cos(bt)+14
The period is 24, so
h(t) = -13cos(π/12 t)+14
To determine the equation for the height of the passenger as a function of time, we can consider the position of the passenger on the Ferris wheel as it rotates. Let's break down the problem step by step:
a) First, let's find the period of the Ferris wheel, which is the time it takes to complete one full rotation. We are given that the Ferris wheel rotates once every 24 seconds.
The period (T) can be calculated using the formula:
T = 2πr/v
Where:
T = period
r = radius
v = velocity
Plugging in the given values:
T = 2π(13 m) / 24 s
Calculating, we get:
T ≈ 8.60 s
Now, we need to find the amplitude (a). The amplitude represents the maximum height the passenger reaches on the Ferris wheel. The amplitude of a Ferris wheel can be determined by finding the difference between the highest and lowest points on the wheel, which is twice the radius.
a = 2r
a = 2(13 m)
Calculating, we get:
a = 26 m
Next, we need to find the angular frequency (b). The angular frequency represents the number of cycles completed per unit of time. It is calculated as the reciprocal of the period.
b = 1/T
Plugging in the previously calculated value for T:
b = 1/8.60 s
Simplifying, we get:
b ≈ 0.116
Finally, to find the vertical shift (d) in the equation, we consider the initial height of the passenger. The passenger gets on at the bottom of the wheel from a ramp, which is one meter above ground level. Therefore, the vertical shift is +1 meter.
d = 1 m
Putting it all together, the equation for the height of the passenger (h) as a function of time (t) is:
h(t) = 26 cos(0.116t) + 1
b) To find the height of the passenger after 55 seconds, we substitute t = 55 into the equation:
h(55) = 26 cos(0.116 * 55) + 1
Using a calculator or computer program to evaluate the trigonometric function, we get:
h(55) ≈ 14.75 meters
Therefore, to the nearest meter, the height of the passenger after 55 seconds is approximately 15 meters.