The blades of a windmill turn on an axis that is 35 feet above the ground. The blades are 10 feet long and complete two rotations every minute. Which of the following equations can be used to model [tex]$h$[/tex], the height in feet of the end of one blade, as a function of time, [tex]$t$[/tex], in seconds? Assume that the blade is pointing to the right, parallel to the ground at [tex]$t=0$[/tex] seconds, and that the windmill turns counterclockwise at a constant rate.
A. [tex]$h=-10 \sin \left(\frac{\pi}{15} t\right)+35$[/tex]
B. [tex]$h=-10 \sin (\pi t)+35$[/tex]
C. [tex]$h=10 \sin \left(\frac{\pi}{15} t\right)+35$[/tex]
D. [tex]$h=10 \sin (\pi t)+35$[/tex]
Answer (2)
The height h of the blade can be modeled as a sinusoidal function: h = A \t \t \t sin ( Bt ) + C .
The amplitude A is the length of the blade: A = 10 .
The vertical shift C is the height of the axis: C = 35 .
The period is 30 seconds, so B = 15 π , giving the equation h = 10 sin ( 15 π t ) + 35 .
Explanation
Problem Analysis We are given that the blades of a windmill turn on an axis that is 35 feet above the ground. The blades are 10 feet long and complete two rotations every minute. We want to model h , the height in feet of the end of one blade, as a function of time, t , in seconds. At t = 0 seconds, the blade is pointing to the right, parallel to the ground, and the windmill turns counterclockwise at a constant rate.
General Equation The height h can be modeled as a sinusoidal function of time t . The general form of the equation is h = A sin ( Bt ) + C , where A is the amplitude, B is related to the period, and C is the vertical shift.
Finding the Amplitude The amplitude A is equal to the length of the blade, which is 10 feet. So, A = 10 .
Finding the Vertical Shift The vertical shift C is equal to the height of the axis above the ground, which is 35 feet. So, C = 35 .
Finding B The period of the rotation is the time it takes to complete one rotation. Since the blades complete two rotations every minute (60 seconds), one rotation takes 30 seconds. Therefore, the period is 30 seconds. We have B = period 2 π = 30 2 π = 15 π .
Determining the Sign Since the blade is pointing to the right at t = 0 and the windmill turns counterclockwise, the height will initially increase. Therefore, we use a positive sine function.
Final Equation Substituting the values of A , B , and C into the general equation, we get h = 10 sin ( 15 π t ) + 35.
Examples
Windmills are used to convert wind energy into electricity. The height of a blade tip can be modeled using trigonometric functions, which helps engineers determine the optimal placement and size of windmills to maximize energy capture. Understanding the sinusoidal motion of the blades allows for efficient design and operation of wind turbines, ensuring a sustainable energy source.
The equation modeling the height of the end of the windmill blade is h = 10 sin ( 15 π t ) + 35 . This corresponds to option C, as it accurately reflects the values for amplitude, vertical shift, and rotation speed of the windmill. Using this model, you can easily calculate the height at any time t .
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