The ferris wheel in an amusement park. It has a diameter of 24m, and it takes 40s to make one complete revolution. If Peter gets on a gondola which is vertically below the centre of the ferris wheel, find his rise in the height after 5s.
I drew a circle, and a vertical radius of 12 m
In 40 s, it rotates 360°, so in 5 sec it rotated 45°
let the vertical position from the centre be y
cos45 = y/12
y = 8.485 m
so he is 12-8.85 or appr 3.5 m high after 5 seconds
To find Peter's rise in height after 5 seconds, we need to calculate the angle turned by the ferris wheel in that time and use trigonometry to determine his vertical displacement.
First, let's find the angle turned by the ferris wheel in 5 seconds. We know that it takes 40 seconds for one complete revolution, so in 1 second, the ferris wheel turns 1/40th of a complete revolution. Thus, in 5 seconds, it turns 5/40th of a complete revolution.
To find the angle turned in radians, we multiply this fraction by 2π since there are 2π radians in one complete revolution. Therefore, the angle turned by the ferris wheel in 5 seconds is:
(5/40) * 2π = (1/8) * 2π = π/4 radians
Now, let's consider the vertical displacement of Peter's gondola. Since he starts below the center of the ferris wheel, the vertical displacement can be found using trigonometry. We will use the fact that the vertical change is equal to the radius of the ferris wheel multiplied by the sine of the angle turned.
Given that the diameter of the ferris wheel is 24m, the radius is half of that, i.e., 12m. Therefore, the vertical displacement of Peter's gondola after 5 seconds is:
Vertical displacement = 12m * sin(π/4)
Using an online calculator or a scientific calculator, we can find that sin(π/4) is approximately equal to 0.7071. Substituting this value, we have:
Vertical displacement = 12m * 0.7071 ≈ 8.4853m
Thus, Peter's rise in height after 5 seconds is approximately 8.4853 meters.