Question 237297: A ferris wheel is 40 meters in diameter and boarded at ground level. The wheel makes one full rotation every 7 minutes, and at time t=0 you are at the 3 o'clock position and ascending. Let f(t) denote your height (in meters) above ground at t minutes. Find a formula for f(t).
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! ferris wheel is 40 meters in diameter.
that means the radius is 20 meters.
the circumference of a circle is given by the equation 2*pi*r
it takes 7 minutes to make a complete circle.
each minute will therefore take you 360/7 degrees around the circle.
that's equal to 51.42857143 degrees
when t = 0, you are at 3 o'clock position which is exactly 20 feet above the ground.
when t = 7/4, you are exactly at the top of the ferris wheel which make you 40 feet above the ground. 7/4 * 360/7 = 360/4 = 90 degrees.
when t = 14/4, you are at the 9 o'clock position which is exactly 20 feet above the ground on the opposite side that you were when t = 0. when t = 14/4, you are at exactly 14/4 * 360/7 = 180 degrees.
when t = 21/4, you are at ground level which winds up being 270 degrees.
when t = 28/4, you are back at the 3 o'clock position because you just made one full turn. 28/4 equals 7 minutes because 7*4 = 28.
bottom line is the ferris wheel is a simulation of your x and y coordinates where the x-axis is at the 9 o'clock and 3 o'clock position, and the y-axis is at the 12 o'clock and 6 o'clock position.
you can simulate where you will be by taking the sin of the angles that you will be at in a specified period of time.
the angle is specified as 360/7 * t which is in minutes.
the elevation is 20 plus the radius of the ferris wheel * the sin of the angle.
the formula would be:
h = f(t) = 20 * (1 + sin(x)) where:
x = 360/7 * t
t = minutes
h = the height you will be at.
the formula says that h is a function of t and is equal to 20 * (1+sin(x)).
let's see how this works.
when t = 0 which is 0 minutes, then:
x = 360/7 * t = 360/7 * 0 = 0 degrees
h = f(t) = 20 * (1 + sin(x)) becomes:
h = f(0) = 20 * (1 + sin(0))
since sin(0) = 0, this becomes:
h = f(t) = 20 * (1 + 0) = 20 * 1 = 20
you are 20 feet above the ground and you are at the 3 o'clock position of the ferris wheel which is half the way to the top.
when t = 7/4 which is 1.75 minutes, then:
x = 360/7 * t = 360/7 * 7/4 = 360/4 = 90 degrees
h = f(t) = 20 * (1 + sin(t)) becomes
h = f(7/4) = 20 * (1 + sin(90))
since sin(90) = 1, this becomes:
h = f(7/4) = 20 * (1 + 1) = 20 * 2 = 40
you are 40 feet above the ground and you are at the 12 o'clock position of the ferris wheel which is all the way to the top.
when t = 14/4 which is 3.5 minutes, then:
x = 360/7 * t = 360/7 * 14/4 = 360*2/4 = 720/4 = 180 degrees
h = f(t) = 20 * (1 + sin(t)) becomes
h = f(14/4) = 20 * (1 + sin(180))
since sin(180) = 0, this becomes:
h = f(14/4) = 20 * (1 + 0) = 20 * 1 = 20
you are 20 feet above the ground and you are at the 9 o'clock position of the ferris wheel which is half the way to the top.
when t = 21/4 which is 5.25 minutes, then:
x = 360/7 * t = 360/7 * 21/4 = 360*3/4 = 1080/4 = 270 degrees
h = f(t) = 20 * (1 + sin(t)) becomes
h = f(21/4) = 20 * (1 + sin(270))
since sin(270) = -1, this becomes:
h = f(14/4) = 20 * (1 - 1) = 20 * 0 = 0
you are 0 feet above the ground and you are at the 6 o'clock position of the ferris wheel which is at the bottom at ground level.
when t = 6 minutes, then:
x = 360/7 * 6 = 2160/7 = 308.5714286 degrees
h = f(t) = 20 * (1 + sin(t)) becomes
h = f(6) = 20 * (1 + sin(308.5714286))
since sin(308.5714286) = -.781831482, this becomes:
h = f(6) = 20 * (1 - .781831482) = 20 * .218168518 = 4.36337351
you are 4.36337351 feet above the ground and you are somewhere between the 4 o'clock and 5 o'clock position on the ferris wheel, much closer to the 5 o'clock position than the 4 o'clock position. This would be roughly 4:43 or a quarter to 5. we are talking about the hour hand position.
looks like the formula works.
if you need a picture, let me know and I'll send you one.
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