A Ferris wheel is 40 meters in diameter and boarded from a platform that is 4 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 2 minutes. How many minutes of the ride are spent higher than 30 meters above the ground?

To find out how many minutes of the ride are spent higher than 30 meters above the ground, we need to first determine how many degrees correspond to a height of 30 meters.

Given that the Ferris wheel has a diameter of 40 meters, we can calculate its radius by dividing the diameter by 2: 40 meters / 2 = 20 meters.

Next, let's calculate the circumference of the Ferris wheel using the formula C = 2πr (circumference equals 2 times pi times radius): C = 2 * 3.14 * 20 meters = 125.6 meters.

Since the wheel completes one full revolution in 2 minutes, we can conclude that in 2 minutes, the wheel covers a circumference of 125.6 meters.

Now, let's determine the proportion of the circumference corresponding to a height higher than 30 meters. To do this, we need to first find the angle that corresponds to a height of 30 meters above the ground.

Using basic trigonometry, we know that the height (30 meters) divided by the radius (20 meters) gives us the sine of the angle corresponding to that height: sin(angle) = 30 meters / 20 meters.

To find the angle itself, we can use the inverse sine function (sin^-1 or arcsin) on both sides of the equation: angle = sin^-1(30 meters / 20 meters).

After calculating the angle, let's determine the proportion of the circumference that corresponds to this angle. To do this, we find the ratio of the angle to 360 (since a full circle has 360 degrees), and then multiply this ratio by the time taken to complete one full revolution (2 minutes).

Using the equation: (angle/360) * 2 minutes, we can calculate the number of minutes spent higher than 30 meters above the ground.

Therefore, the number of minutes of the ride spent higher than 30 meters above the ground will be:

(angle/360) * 2 minutes

To solve this problem, we need to determine the position of the Ferris wheel as it rotates and calculate the time it spends above 30 meters above the ground.

First, let's find the total distance traveled by the Ferris wheel in one full revolution:
Circumference of the Ferris wheel = π * diameter
Circumference = π * 40 meters = 40π meters

Next, let's find the time it takes for the Ferris wheel to complete one full revolution:
Time for one full revolution = 2 minutes

Now, we need to calculate the height of the Ferris wheel at the 6 o'clock position:
Height of the Ferris wheel at the 6 o'clock position = diameter / 2 + platform height
Height = 40 meters / 2 + 4 meters = 20 meters + 4 meters = 24 meters

We can now determine the angle through which the Ferris wheel rotates to reach 30 meters above the ground:
Angle = arccos(30 meters / 24 meters)
Angle = arccos(5/4)
Angle ≈ 0.92 radians

Finally, we can calculate the time spent above 30 meters by finding the fraction of the total angle traversed by the Ferris wheel and multiplying it by the time for one full revolution:
Time spent above 30 meters = Angle / (2π) * Time for one full revolution
Time = 0.92 / (2π) * 2 = 0.92 / π ≈ 0.2933 minutes

Therefore, approximately 0.2933 minutes of the ride are spent higher than 30 meters above the ground.

let's use a sine function ....

amplitude = 20
period = 2π/k = 2
2k = 2π
k = π

so far we have
height = 20 sin π(t + c) + d , assuming we have a phase shift and a vertical shift
clearly, the whole sine curve has to shifted upwards by 24 units, so that the normal min of -20 becomes a min of 4
so we have
height = 20 sin π(t+c) + 24

1. when t = 0 , we want height to be 4
2. when t = 1 , we want the height to be 44

1.
4 = 20sin π(c) + 24
-20 = 20sin πc
sin πc = -1
but we know that sin 3π/2 = -1
then cπ = 3π/2
c = 3/2

equation is
height = 20 sin π(t + 3/2) + 24

check for t = 1, (should get 44)
height = 20 sin π(1 + 3/2) + 24
= 20 sin 5π/2 + 24 = 20(1) + 24 = 44

all looks good.

so we want h ≥ 30
20 sin π(t+3/2) + 24 = 30
sin π(t+3/2) = 6/20 = 3/10 = .3
setting my calculator to radians , I get
π(t+3/2) = .3047 or π-.3047 = 2.837
t + 1.5 = .3047/π or t+1.5 = 2.837/π
t = -1.403 or t = -.597

BUT, we should have positive times,
since the period of the curve is 2, adding 2 to each of these times will also satisfy the equation
so t = -1.403+2 = .597
and t = -.597 + 2 = 1.403

so for the time from t = .597 to t = 1.403 the height will be ≥ 30

So it will be above 30 for( 1.403-.597) minutes
= .806 minutes or appr 48.4 seconds