Multiple Choice
On the unit circle, what relationship do the ratios and share for any real values of and ?
3. Unit Circle
Trigonometric Functions on the Unit Circle
Multiple Choice
Which of the following situations can be modeled with a periodic function?
A
The height of a point on a Ferris wheel as it rotates over time
B
The temperature in a city that steadily increases throughout the year without repeating
C
The amount of water in a tank that is being filled at a constant rate
Verified step by step guidance1
Understand the definition of a periodic function: a function \( f(t) \) is periodic if there exists a positive number \( P \) such that \( f(t + P) = f(t) \) for all values of \( t \). This means the function repeats its values in regular intervals.
Analyze the first situation: the height of a point on a Ferris wheel as it rotates over time. Since the Ferris wheel rotates continuously, the height of the point repeats after each full rotation, making the height a periodic function of time.
Analyze the second situation: the temperature in a city that steadily increases throughout the year without repeating. Since the temperature does not repeat values in a regular cycle but instead increases steadily, it cannot be modeled by a periodic function.
Analyze the third situation: the amount of water in a tank being filled at a constant rate. This amount increases continuously without repeating, so it is not periodic.
Conclude that only the first situation, the height of a point on a Ferris wheel as it rotates, can be modeled with a periodic function because it exhibits regular, repeating behavior over time.
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