The height of a rider on a Ferris wheel can be modeled by...

The height of a rider on a Ferris wheel can be modeled by the function $h=k-A \cos (\omega t),$ where $t$ is measured in minutes and $h$ in feet. If the height of the rider alternates between 10 feet and 80 feet and the rider completes one revolution every 4 minutes, find a formula for $h.$ a. $h=45-40 \cos (8 \pi t)$ b. $h=40-45 \cos (4 t)$ c. $h=45-35 \cos \left(\frac{\pi}{2} t\right)$ d. $h=35-45 \cos \left(\frac{\pi}{2} t\right)$

The height of a rider on a Ferris wheel can be modeled by the function $h=k-A \cos (\omega t),$ where $t$ is measured in minutes and $h$ in feet. If the height of the rider alternates between 10 feet and 80 feet and the rider completes one revolution every 4 minutes, find a formula for $h.$
a. $h=45-40 \cos (8 \pi t)$
b. $h=40-45 \cos (4 t)$
c. $h=45-35 \cos \left(\frac{\pi}{2} t\right)$
d. $h=35-45 \cos \left(\frac{\pi}{2} t\right)$

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Key Concepts

Amplitude

Amplitude is the measure of how far the function's values deviate from its central value, representing half the distance between the maximum and minimum values of a periodic function. It determines the vertical stretch or compression of the graph.

Vertical Shift

The vertical shift moves the entire graph up or down, setting the midline of the function. It is calculated as the average of the maximum and minimum values and represents the central value around which the function oscillates.

Period

The period is the duration required for the function to complete one full cycle of its oscillatory behavior. It is related to the angular frequency by the formula period = (2?)/(angular frequency), defining the scale of the horizontal axis.

Angular Frequency

Angular frequency indicates how rapidly the function cycles and is a factor that multiplies the time variable in trigonometric functions. It is directly related to the period of the function and adjusts the rate of oscillation.

Trigonometric Modeling of Periodic Phenomena

Trigonometric modeling involves using functions like cosine or sine, modified by amplitude, vertical shift, and angular frequency, to accurately represent periodic behavior in various contexts. This technique translates real-world cyclical processes into mathematical models.

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