so what fraction of the circle circumference is 18 - 3 = 15 meters above the bottom of the circle?
R = 25/2 = 12.5 meters
15 - 12.5 = 2.5 meters above circle center
A = angle above circle center
sin A = 2.5 / 12.5
A = 11.54 degrees
2 A = 23.07 degrees
fraction above = [360 - 23.07-180 ] / 360= 0.436
0.436 * 4 minutes = 1.743minutes
A Ferris wheel is 25 meters in diameter and boarded from a platform that is 3 meters above the ground. The six o'clock position on the Ferris wheel is level with the loading platform. The wheel completes 1 full revolution in 4 minutes. How many minutes of the ride are spent higher than 18 meters above the ground?
3 answers
a = 25/2 = 12.5
period = 2π/k
k = 2π/4 = π/2
so let's start with height = 12.5sin (π/2 t)
this would give us a minimum of -12.5 and a max of 12.5, but we want the
minimum to be 3, so we have to raise our curve 15.5
so next adjustment:
height = 12.5sin(πt/2) + 15.5
but, a rough sketch shows that our min of 3 is at t = -1 , to we have to move
our curve 1 unit to the right:
Final equation: height = 12.5 sin(π/2 (t-1)) + 15.5
( I graphed this using DESMOS, and it is correct)
we want 12.5 sin(π/2 (t-1)) + 15.5 ≥ 18
let's consider 12.5 sin(π/2 (t-1)) + 15.5 = 18
12.5sin(π/2 (t-1)) = 2.5
sin(π/2 (t-1)) = .2
Calculator time , set to radians :
π/2 (t-1) = .201358 or π/2 (t-1) = π - .201358
t-1 = .128188 or t-1 = 1.8718
t = 1.128 or t = 2.8718
between those two times, the rider is ≥ 18 m
(confirmed by DESMOS graph)
So the rider spends 2.8718-1.128 or 1.744 minutes above 18 m.
period = 2π/k
k = 2π/4 = π/2
so let's start with height = 12.5sin (π/2 t)
this would give us a minimum of -12.5 and a max of 12.5, but we want the
minimum to be 3, so we have to raise our curve 15.5
so next adjustment:
height = 12.5sin(πt/2) + 15.5
but, a rough sketch shows that our min of 3 is at t = -1 , to we have to move
our curve 1 unit to the right:
Final equation: height = 12.5 sin(π/2 (t-1)) + 15.5
( I graphed this using DESMOS, and it is correct)
we want 12.5 sin(π/2 (t-1)) + 15.5 ≥ 18
let's consider 12.5 sin(π/2 (t-1)) + 15.5 = 18
12.5sin(π/2 (t-1)) = 2.5
sin(π/2 (t-1)) = .2
Calculator time , set to radians :
π/2 (t-1) = .201358 or π/2 (t-1) = π - .201358
t-1 = .128188 or t-1 = 1.8718
t = 1.128 or t = 2.8718
between those two times, the rider is ≥ 18 m
(confirmed by DESMOS graph)
So the rider spends 2.8718-1.128 or 1.744 minutes above 18 m.
good one