As you ride a ferris wheel, your distance from the ground varies sinusoidally with time. You enter the ferris wheel from the bottom, which is 8 feet from the ground. The ferris wheel has a diameter of 60 feet and makes a full revolution every minute. What is your height above the ground at t = 20 seconds?
wheel has a diameter of 60 feet
Start with y = 30cos(t)
You enter the Ferris wheel from the bottom, which is 8 feet from the ground.
y = 38 - 30cos(t)
a full revolution every minute
so, the period is 1, and you have
y = 38 - 30cos(2πt)
So now plug in t = 1/3
To find your height above the ground at t = 20 seconds, we can use the equation for a sinusoidal function. The general equation for a sinusoidal function is:
y = A * sin(B * (x - C)) + D
where:
A = amplitude
B = period (time for one complete cycle)
C = horizontal shift (phase shift)
D = vertical shift
In this case, the amplitude is half the diameter of the ferris wheel, which is 60 / 2 = 30 feet. The period is the time for one complete revolution, which is 1 minute or 60 seconds. The horizontal shift is determined by the time it takes to complete one revolution, so C = 0. The vertical shift is the height of the bottom of the ferris wheel, which is 8 feet.
Plugging these values into the equation, we have:
y = 30 * sin((2π / 60) * (t - 0)) + 8
Now, we can substitute t = 20 seconds to find the height above the ground:
y = 30 * sin((2π / 60) * (20 - 0)) + 8
Simplifying further:
y = 30 * sin(2π / 3) + 8
Using a calculator, we can find the value of sin(2π / 3) ≈ 0.866:
y ≈ 30 * 0.866 + 8
y ≈ 25.98 + 8
y ≈ 33.98
So, at t = 20 seconds, your height above the ground is approximately 33.98 feet.