A Ferris wheel is 25 meters in diameter and boarded from a platform that is 1 meter above the ground. The six o’clock position on the Ferris wheel is level with the loading platform. The wheel completes 1 full revolution in 10 minutes. The function h (t) gives a person’s height in meters above the ground t minutes after the wheel begins to turn.
Find the amplitude, midline, and period of h (t).
Find a formula for the height function h (t).
How high off the ground is a person after 5 minutes?
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To find the amplitude, midline, and period of the height function h(t), we can analyze the given information:
Amplitude: The amplitude of the function represents the maximum vertical distance from the midline to the highest or lowest point on the graph. In this case, the Ferris wheel reaches its highest point when it is 25 meters in diameter, so the amplitude is half that value: 25 / 2 = 12.5 meters.
Midline: The midline is the horizontal line that represents the average value of the function. Since the Ferris wheel is initially boarded from a platform that is 1 meter above the ground, the midline is at a height of 1 meter.
Period: The period of the function is the time it takes for one complete cycle to occur. In this case, the Ferris wheel completes one revolution in 10 minutes, so the period is 10 minutes.
Therefore, the amplitude is 12.5 meters, the midline is 1 meter, and the period is 10 minutes.
To find a formula for the height function h(t), we can use the equation for a sine function:
h(t) = A * sin(B(t - C)) + D
where:
A = amplitude
B = frequency (2π / period)
C = horizontal shift
D = vertical shift (midline)
Using the values we found earlier, we get:
h(t) = 12.5 * sin((2π / 10)(t - C)) + 1
Since the six o'clock position (when the Ferris wheel starts) corresponds to t = 0, we need to find the value of C. Since the function is at its highest point (amplitude) at t = 0, we can deduce that C = 0.
Therefore, the formula for the height function is:
h(t) = 12.5 * sin((2π / 10)t) + 1
To find the height off the ground after 5 minutes, we can substitute t = 5 into the height function:
h(5) = 12.5 * sin((2π / 10) * 5) + 1
= 12.5 * sin(π) + 1
= 12.5 * 0 + 1
= 1
So, a person is 1 meter above the ground after 5 minutes.
To find the amplitude, midline, and period of the height function h(t), we can refer to the given information about the Ferris wheel.
1. Amplitude: The amplitude of the function represents the maximum vertical distance from the midline. Since the Ferris wheel has a diameter of 25 meters, the radius (and thus the amplitude) is half of that, which is 12.5 meters.
2. Midline: The midline is the horizontal line that represents the average height. In this case, the midline is 1 meter above the ground since the loading platform is 1 meter above the ground.
3. Period: The period of a periodic function is the time it takes to complete one full cycle. In this case, the Ferris wheel completes one full revolution in 10 minutes, which gives us the period of the function as 10 minutes.
Therefore, the amplitude is 12.5 meters, the midline is 1 meter, and the period is 10 minutes.
Now, let's find a formula for the height function h(t). Since the Ferris wheel completes one revolution every 10 minutes, its angular frequency (2π divided by the period) is 2π/10 = π/5.
The general formula for a sinusoidal function is given by h(t) = A * sin(ωt) + B, where A is the amplitude, ω is the angular frequency, t is the time, and B is the midline.
Substituting the values we found, we get the height function as h(t) = 12.5 * sin((π/5)t) + 1.
To find how high off the ground a person is after 5 minutes, we substitute t = 5 into the height function:
h(5) = 12.5 * sin((π/5) * 5) + 1
= 12.5 * sin(π) + 1
= 12.5 * 0 + 1
= 1
Therefore, a person is 1 meter above the ground after 5 minutes.
Hint: sketch a picture. Use the fact a complete revolution takes 10 minutes... then the function at time 0, that is f(0) = how high off the ground...
Amplitude = wheel radius
Period = how long for one revolution
That should start you off nicely...