A circular Ferris wheel has a radius of 9 meters. The ride rotates a rate of 10 degrees per second. When you get in the ride at the bottom the seat is 2 meters above the ground at its lowest point in meters. How high is the seat after 54 seconds.
To determine the height of the seat after 54 seconds, we need to find the angle of rotation at that time and calculate the corresponding height.
The Ferris wheel rotates at a rate of 10 degrees per second. Therefore, after 54 seconds, the total angle of rotation would be:
54 seconds * 10 degrees/second = 540 degrees
Since the Ferris wheel is circular and has a radius of 9 meters, we can consider it as a circle where the angle is measured in radians. To convert degrees to radians, we use the conversion factor:
1 radian = (180 degrees)/π
Therefore, the angle in radians is:
540 degrees * (π/180 degrees) = 9π radians
Now, we can calculate the height of the seat above the ground. At the lowest point, the seat is 2 meters above the ground, which we will call the reference height.
The height of the seat at any given angle can be determined by using trigonometry on the right triangle formed between the center of the Ferris wheel, the reference height, and the seat height.
Given that the radius of the Ferris wheel is 9 meters and the reference height is 2 meters, we can use the sine function as follows:
sin(angle) = (opposite-side)/(hypotenuse)
sin(angle) = (height)/(radius)
Solving for the height:
height = sin(angle) * radius
height = sin(9π) * 9
Now we can calculate the height:
height = sin(9π) * 9 = -9
Therefore, the height of the seat after 54 seconds is -9 meters.
Note: The negative sign indicates that the seat is below the reference height at that moment.