The equation $h=7 \sin \left(\frac{\pi}{21} t\right)+28$ can be used to model the height, $h$, in feet of the end of one blade of a windmill turning on an axis above the ground as a function of time, $t$, in seconds. How long is the blade? Assume that the blade is pointing to the right, parallel to the ground, at $t=0$, and that the windmill turns counterclockwise at a constant rate.
A. 7 feet
B. 14 feet
C. 21 feet
D. 28 feet
Answer (1)
The height of the center of the windmill is 28 feet.
The amplitude of the sinusoidal function is 7 feet, which represents the radius of the circle traced by the end of the blade.
The length of the blade is twice the radius.
The length of the blade is $2 \times 7 =
14 f ee t
Explanation
Understanding the Problem The height of the end of a windmill blade is modeled by the equation h = 7 sin ( 21 π t ) + 28 , where h is the height in feet and t is the time in seconds. We want to find the length of the blade.
Finding the Radius The height of the center of the windmill is given by the constant term in the equation, which is 28 feet. The amplitude of the sinusoidal function is 7 feet. The amplitude represents the radius of the circle traced by the end of the blade.
Calculating the Length The length of the blade is twice the radius. Therefore, the length of the blade is 2 × 7 = 14 feet.
Examples
Understanding sinusoidal functions like the one in this problem can help model various real-world phenomena, such as the motion of a pendulum, the oscillation of a spring, or the fluctuations in temperature over time. In this case, knowing the height of the windmill blade helps engineers design and maintain these structures safely. By analyzing the equation, we can determine the maximum and minimum heights, ensuring that the windmill operates efficiently and doesn't collide with any surrounding objects. This also helps in optimizing the placement of windmills in wind farms to maximize energy generation.
