Amare wants to ride a Ferris wheel that sits four meters above and has a diameter of 50 meters. It takes 6 minutes to do three r
evolutions on the Ferris wheel. Complete the function, h(t), which models Amare’s height above the ground, in meters, as a function of time, t, in minutes. Assume he enters the ride at the low point when t=0
To answer this question, we begin to transform the speed of revolutions per minute in radians per minute. They tell us that the speed of the wheel is 3 turns in 6 minutes. So: 1 revolution = 2π (2π / 1 turn) * (3 turns / 6min) = 6π/ 6min = 1π / min. The wheel turns π or 180 ° in one minute. We already have the angular velocity w. We know that the lowest point of the wheel is 4 meters above the ground and that it returns to the same point every 2π Therefore, the function sought is periodic and must be equal to 4 for allvalues of time k, where k is an even number 2, 4, 6, 8, .., k
Then the function must have the form rsin(wt) where "t" is the elapsed time, "w" is the previously calculated angular velocity, and "r" is the radius of the wheel. The minimum value of the function must be 4 and the maximum value 54. Therefore, the function is:
h (t) = 4 + 25 + 25sin (π×t + 3π / 2) Where 3π/ 2 is the phase angle, which indicates that the movement starts at the instant t = 0 at the lowest point of the wheel that equals 3π / 2. You can verify the answer in the following way:
After 1 minute, the wheel should have rotated 180 ° or π. Therefore, the person must be at the highest point of the wheel and his height must be 54 m. When you replace t = 1 in the formula, you get h = 54m After 2 minutes, the wheel should have rotated 360 ° or 2π. Therefore, the person must be at the starting point and their height must be 4 m. By replacing t = 2 in the formula you will get h = 4m After 0.5 minutes, the wheel should have rotated 90 ° or π / 2. Therefore, the person must be in the right half of the wheel and his height must be 29 m. When replacing t = 0.5 in the formula you will get h = 29m
