A physical therapy center has a bicycle that patients use for exercise. The height, in inches (in), of thebicycle pedal above level ground periodically increases and decreases when used. The figure gives theposition of the pedal P at a height of 12 inches above the ground at time t = 0 seconds. The pedal’s 8-incharm defines the circular motion of the pedal. If a patient pedals 1 revolution per second, which of thefollowing could be an expression for h t( ), the height, in inches, of the bicycle pedal above level ground attime t seconds?(A) 8 1− 2sin t(B) 12 − 8sin t(C) 8 1− 2sin( )2 t(D) 12 − 8sin( )2 t
Question
A physical therapy center has a bicycle that patients use for exercise. The height, in inches (in), of thebicycle pedal above level ground periodically increases and decreases when used. The figure gives theposition of the pedal P at a height of 12 inches above the ground at time t = 0 seconds. The pedal’s 8-incharm defines the circular motion of the pedal. If a patient pedals 1 revolution per second, which of thefollowing could be an expression for h t( ), the height, in inches, of the bicycle pedal above level ground attime t seconds?(A) 8 1− 2sin t(B) 12 − 8sin t(C) 8 1− 2sin( )2 t(D) 12 − 8sin( )2 t
Solution
The height of the bicycle pedal above the ground can be modeled as a sinusoidal function because it periodically increases and decreases. The amplitude of this function is the radius of the circular motion, which is 8 inches. This is the maximum deviation from the mean height, which is 12 inches.
The function also completes one full cycle every second, which means the frequency is 1. Therefore, the function can be written in the form h(t) = A + Bsin(Ct), where A is the mean height, B is the amplitude, and C is the frequency.
Substituting the given values into this equation gives h(t) = 12 + 8sin(t). However, because the pedal is at its maximum height at t = 0, and the sine function starts at 0, we need to shift the sine function by π/2 to the left. This gives h(t) = 12 - 8sin(t).
Therefore, the correct answer is (B) 12 - 8sin(t).
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