Suppose that you are sitting in a chair on a Ferris wheel that has a diameter of 120 feet that is boarded 6 feet above the ground at the 6 o’clock position. If the Ferris wheel moves in a counterclockwise direction and makes one revolution every 12 minutes, how high will your chair be off the ground after 6 minutes? After 32 minutes? At 9 minutes, is your chair going up or
going down?
To solve this problem, we can use the information given about the Ferris wheel's diameter, rotation speed, and initial position.
First, let's calculate the circumference of the Ferris wheel. Since the diameter is given as 120 feet, the radius is half of that, which is 60 feet. The formula for the circumference of a circle is C = 2πr, where π is a mathematical constant approximately equal to 3.14. Plugging in the values, we get C = 2π(60) = 120π feet.
Next, we can determine how much the Ferris wheel rotates in 6 minutes. If the Ferris wheel makes one revolution (360 degrees) every 12 minutes, in 6 minutes it will rotate halfway, which is 180 degrees.
Now, we need to find out where your chair will be after 6 minutes. Since your chair starts at the 6 o'clock position, and we know the Ferris wheel has traveled 180 degrees counterclockwise, your chair will be at the 12 o'clock position.
To find out how high your chair is off the ground, we need to know the height of the 12 o'clock position. Since the Ferris wheel is boarded 6 feet above the ground at the 6 o'clock position, we can assume it will be the same at the 12 o'clock position.
Therefore, after 6 minutes, your chair will be 6 feet above the ground.
To calculate how high your chair will be off the ground after 32 minutes, we need to determine how many revolutions the Ferris wheel has made in that time. Since one revolution takes 12 minutes, in 32 minutes, the Ferris wheel completes 2 full revolutions (24 minutes), plus an additional 8 minutes. This means it rotates 720 degrees (2 revolutions of 360 degrees each).
Again, we need to find the position of your chair after 720 degrees. Since your chair starts at the 6 o'clock position and the Ferris wheel has traveled a full circle (360 degrees), your chair will be back at the 6 o'clock position.
Therefore, after 32 minutes, your chair will be 6 feet above the ground.
To determine whether your chair is going up or down at the 9-minute mark, we need to calculate the position after 9 minutes. Since the Ferris wheel rotates one full revolution (360 degrees) every 12 minutes, at the 9-minute mark, it will have rotated 3/4 of a revolution (270 degrees).
To find the position of your chair, we need to add 270 degrees to the starting position of 6 o'clock. Adding 270 degrees to 180 degrees (the starting position of your chair after 6 minutes) gives us 450 degrees.
To determine whether your chair is going up or down, we need to look at the angle relative to the highest point (12 o'clock position). At 450 degrees, your chair will be above the highest point but still moving up.
Therefore, at 9 minutes, your chair is going up.
boarding at t=0, the height y at time t minutes is
y = -60cos(pi/6 t) + 6
Now just start plugging in your numbers.