center of wheel is 36m high ... (70 / 2) + 1
how much time >8m above center?
draw a sketch ... diameter is horizontal at 36m
... 8m perpendicular from diameter to edge of wheel
... sine of central angle is ... 8 / 35
180º plus twice central angle is portion of circumference BELOW 44m
A Ferris wheel is boarding platform is 1 meters above the ground, has a diameter of 70 meters, and rotates once every 5 minutes.
How many minutes of the ride are spent higher than 44 meters above the ground?
2 answers
Or , using strictly trig and a sine curve:
period : 2π/k = 5
k = 2π/5
possible equation:
y = 35sin (2π/5 t) + 36
we want our curve to be such that, when t = 0, y = 1, so we need a phase shift.
y = 35sin 2π/5(t + d) + 36
when t=0,y=1
35sin 2π/5(0 + d) + 36 = 1
35sin 2π/5(d) = -35
sin 2π/5d = -1
we know sin 3π/2 = -1
2π/5d = 3π/2
d = 15/4
y = 35sin 2π/5(t + 15/4) + 36
35sin 2π/5(t + 15/4) + 36 > 44
sin 2π/5(t + 15/4) > 8/35
let's look when sin 2π/5(t + 15/4) = 8/35
2π/5(t+15/4) = .23061.. or 2π/5(t+15/4) = π - .23061 = 2.91098
t+15/4 = .183514.... or t+15/4 = 2.316485
t = -3.5665 or t = -1.4335
but the period of our curve is 5 , let's add 5 to our answers to get more
so t = 1.4335 or t = 3.5665
then we are above 44 m from time 1.4335 min to 3.5665 min or for
a period of 2.133 minutes
confirmation:
https://www.wolframalpha.com/input/?i=plot+sin+(2%CF%80%2F5(t+%2B+15%2F4))++%3D+8%2F35+for+t+%3D+0+to+8
period : 2π/k = 5
k = 2π/5
possible equation:
y = 35sin (2π/5 t) + 36
we want our curve to be such that, when t = 0, y = 1, so we need a phase shift.
y = 35sin 2π/5(t + d) + 36
when t=0,y=1
35sin 2π/5(0 + d) + 36 = 1
35sin 2π/5(d) = -35
sin 2π/5d = -1
we know sin 3π/2 = -1
2π/5d = 3π/2
d = 15/4
y = 35sin 2π/5(t + 15/4) + 36
35sin 2π/5(t + 15/4) + 36 > 44
sin 2π/5(t + 15/4) > 8/35
let's look when sin 2π/5(t + 15/4) = 8/35
2π/5(t+15/4) = .23061.. or 2π/5(t+15/4) = π - .23061 = 2.91098
t+15/4 = .183514.... or t+15/4 = 2.316485
t = -3.5665 or t = -1.4335
but the period of our curve is 5 , let's add 5 to our answers to get more
so t = 1.4335 or t = 3.5665
then we are above 44 m from time 1.4335 min to 3.5665 min or for
a period of 2.133 minutes
confirmation:
https://www.wolframalpha.com/input/?i=plot+sin+(2%CF%80%2F5(t+%2B+15%2F4))++%3D+8%2F35+for+t+%3D+0+to+8
1 answer