To model the hamster's distance from the ground as a function of time, we need to consider the motion of the hamster in the circular wheel.
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Determine the radius of the wheel:
- The diameter of the wheel is 8 cm, so the radius \( r \) is: \[ r = \frac{8 \text{ cm}}{2} = 4 \text{ cm} \]
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Determine the height of the center of the wheel from the ground:
- The wheel is 2 cm off the ground, so the center of the wheel is at: \[ \text{height of center} = 2 \text{ cm} + 4 \text{ cm} = 6 \text{ cm} \] (Here, we add the radius to the height of the base of the wheel to find the height of the center of the wheel).
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Determine the angular frequency:
- The hamster makes 4 rotations per second. In terms of radians, one rotation corresponds to \( 2\pi \) radians. Therefore, the angular frequency \( \omega \) in radians per second is: \[ \omega = 4 \text{ rotations/sec} \cdot 2\pi \text{ radians/rotation} = 8\pi \text{ radians/sec} \]
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Model the vertical motion of the hamster:
- The vertical position of the hamster at any time \( t \) can be modeled with a cosine function, where the cosine function will represent the periodic motion due to its circular path. Since at \( t = 0 \) the hamster is at the bottom of the wheel (lowest point, 2 cm above the ground), the function can be modeled as follows: \[ h(t) = 6 - 4\cos(8\pi t) \] Here, \( 6 \) represents the height of the center of the wheel, \( 4 \) (the radius) determines the amplitude of the oscillation, and \( \cos(8\pi t) \) captures the periodic motion of the hamster.
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Final model:
- The distance from the ground can therefore be modeled by the equation: \[ h(t) = 6 - 4\cos(8\pi t) \] This equation gives the height of the hamster from the ground as a function of time \( t \) in seconds.
Thus, the final equation modeling the hamster's distance from the ground in terms of time is: \[ h(t) = 6 - 4\cos(8\pi t) \]