The height, [tex]$h$[/tex], in feet of the tip of the hour hand of a wall clock varies from 9 feet to 10 feet. Which of the following equations can be used to model the height as a function of time, [tex]$t$[/tex], in hours? Assume that the time at [tex]$t=0$[/tex] is 12:00 a.m.
[tex]$h=0.5 \cos \left(\frac{\pi}{12} t\right)+9.5$[/tex]
[tex]$h=0.5 \cos \left(\frac{\pi}{6} t\right)+9.5$[/tex]
[tex]$h=\cos \left(\frac{\pi}{12} t\right)+\theta$[/tex]
[tex]$h=\cos \left(\frac{\pi}{6} t\right)+\theta$[/tex]
Answer (2)
The amplitude is calculated as half the difference between the maximum and minimum heights: 2 10 − 9 = 0.5 .
The vertical shift is the average of the maximum and minimum heights: 2 10 + 9 = 9.5 .
The angular frequency is calculated as ω = 12 2 π = 6 π .
The equation that models the height as a function of time is: h = 0.5 cos ( 6 π t ) + 9.5 .
Explanation
Understanding the Problem We are given that the height, h , in feet of the tip of the hour hand of a wall clock varies from 9 feet to 10 feet. We need to find an equation that models the height as a function of time, t , in hours. The time at t = 0 is 12.00 a.m.
Key Observations The height varies between 9 and 10 feet, which suggests a sinusoidal function (cosine or sine) with a vertical shift. The hour hand completes a full cycle in 12 hours.
Calculating Amplitude The amplitude of the cosine function is half the difference between the maximum and minimum heights: 2 10 − 9 = 0.5
Calculating Vertical Shift The vertical shift is the average of the maximum and minimum heights: 2 10 + 9 = 9.5
Calculating Angular Frequency The hour hand completes a full cycle in 12 hours, so the period is 12. The angular frequency ω is given by: ω = T 2 π = 12 2 π = 6 π
General Cosine Function The general form of the cosine function is: h = A cos ( ω t ) + D where A is the amplitude, ω is the angular frequency, and D is the vertical shift.
Substituting Values Substituting the values we found: h = 0.5 cos ( 6 π t ) + 9.5
Final Equation Comparing the derived equation with the given options, we find that the correct equation is: h = 0.5 cos ( 6 π t ) + 9.5
Examples
Imagine you're tracking the height of a Ferris wheel seat as it rotates. The height oscillates between a minimum and maximum value, just like the hour hand on a clock. By modeling this height with a cosine function, you can predict the seat's height at any given time. This is useful for safety systems, timing rides, or even designing the structure of the Ferris wheel itself. Understanding sinusoidal functions helps in analyzing any periodic motion, from sound waves to planetary orbits.
The correct equation that models the height of the hour hand as a function of time is h = 0.5 cos ( 6 π t ) + 9.5 . This equation accounts for the amplitude, vertical shift, and angular frequency based on the height's variation. It corresponds to the option provided in the question.
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