A Ferris wheel has a radius of 8 m and rotates once every 22 seconds. Suppose a passenger boards the Ferris wheel at the lowest point which is 2 m above the ground. If the ride begins once this passenger boards the lowest car, determine time at which this passenger's height above the ground first reaches 10 m.
Round your answer correct to two decimal places. Enter values such as: 0.02, 4.00, 5.30, 9.24
0.08
the hard way: solve
10-8cos(π/11 x) = 10
the easy way: when the car is at height 10m, it has gone 1/4 turn, so it will take 22/4 = 5.5 seconds
The height of a person on a Ferris wheel is described by a sinusoidal function, that is, either by a sine or a cosine curve.
since the radius is 8 m, and you want the minimum to be 2m
we have a minimum of 2
a maximum of 18
since the period is 22 seconds, we have
a min of 2 when t = 0
a height of 10 when t = 22/4 or 5.5 second
a height of 18 when t = 22/2 or 11 seconds
a height of 10 when t = 3(22/4) or 16.5 seconds, (on its way down)
a height of 2 when t = 22 seconds
btw, the equation would be
height = 8sin ( π/11(t - 11/2) + 10
Proof: (copy and paste this URL)
www.wolframalpha.com/input/?i=y+%3D+8sin+%28+%CF%80%2F11%28x+-+11%2F2%29%29+%2B+10
Notice that oobleck and I have different equations, but they represent the
same curve, and we have the same result
(change my equation in the Wolfram link to oobleck's and you will get the
same trig curve.)
To determine the time at which the passenger's height above the ground first reaches 10 m, we need to analyze the motion of the Ferris wheel.
The Ferris wheel's motion can be described as circular motion with a radius of 8 m and a period of 22 seconds. The height of the passenger above the ground at any given time can be calculated using trigonometry.
First, let's find the equation for the passenger's height above the ground as a function of time. We can use the equation:
h(t) = r + R * sin(2πt / T)
Where r is the initial height (2 m), R is the radius of the Ferris wheel (8 m), t is the time, and T is the period of the Ferris wheel's rotation (22 seconds).
Now, we need to find the time at which the passenger's height first reaches 10 m. We can set up the equation:
10 = 2 + 8 * sin(2πt / 22)
To solve for t, we'll isolate the sine function:
8 * sin(2πt / 22) = 10 - 2
8 * sin(2πt / 22) = 8
Next, divide both sides by 8:
sin(2πt / 22) = 1
Now we solve for t by taking the inverse sine (arcsine) of both sides:
2πt / 22 = arcsin(1)
arcsin(1) is equal to π/2, so we have:
2πt / 22 = π/2
Now solve for t:
t = (π/2) * (22 / 2π)
t = 11
Therefore, the passenger's height above the ground first reaches 10 m at t = 11 seconds.
So, the time at which this passenger's height above the ground first reaches 10 m is 11.00 seconds.