Answer :
The equation \( h = -150 \sin(\theta) + 260 \) describes the motion of a point on the end of a blade of a wind turbine in terms of its height \( h \), in feet, above the ground as a function of the angle of rotation \( \theta \) of the blade. We can break down this trigonometric function to understand the movement of the turbine's blade:
1. **Amplitude**: The coefficient of the sine function, \( -150 \), indicates the amplitude of the motion. In this context, the amplitude represents half the total vertical movement of the blade tip above and below the central position. However, the fact that it is negative means that the sine function is inverted. So, if we ignore the negative sign, the amplitude is 150 feet, meaning that as the blade rotates, the tip moves 150 feet up and down from its central height.
2. **Mid-Height**: The constant term \( +260 \) in the equation gives us the mid-height of the blade's motion. It is the vertical position around which the sine function oscillates. In this case, the central height is 260 feet above the ground.
3. **Maximum and Minimum Heights**: Knowing the amplitude and the mid-height, we can find the maximum and minimum heights of the blade's tip. The maximum height of the blade's tip will be when \( \sin(\theta) \) is at its minimum value, which is -1 due to the negative amplitude. So the maximum height is \( 260 - 150 = 110 \) feet. The minimum height will be when \( \sin(\theta) \) is at its maximum value, which is 1. So the minimum height is \( 260 + 150 = 410 \) feet. These are inverted due to the negative sign in front of the amplitude.
4. **Direction of Rotation**: Since \( \sin(\theta) \) starts from 0 when \( \theta = 0 \) and the amplitude is negative, the point on the blade starts directly to the right of the wind turbine and moves downward as it begins to rotate. If you were standing above the turbine looking down, the blades would appear to rotate clockwise due to this initial downward movement.
5. **Period of Motion**: In trigonometric functions, the standard period of a sine function is \( 2\pi \) radians, which corresponds to one complete rotation of 360 degrees. Since there is no coefficient inside the sine function that affects \( \theta \), the point on the blade completes one full cycle of up and down motion for every 360 degrees of rotation. There is also no phase shift since the sine function's argument is simply \( \theta \).
So, in conclusion, this wind turbine has blades that, when observed from the side, move in a sinusoidal pattern between a maximum height of 410 feet and a minimum height of 110 feet above the ground. The motion begins with the blade tip moving downward from the 260-feet mid-height position as the blades rotate. From an overhead view, the wind turbine appears to be spinning in a clockwise direction.
