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?

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

To find out how many minutes of the ride are spent higher than 44 meters above the ground, we need to determine the time it takes for the Ferris wheel to reach that height.

1. First, we need to find the radius of the Ferris wheel. The diameter is given as 70 meters, so the radius (r) is half of that: 70 meters / 2 = 35 meters.

2. We can use the formula for the height (h) of an object moving in a circular path: h = r - r*cos(theta), where r is the radius and theta is the angle of rotation.

3. In this case, we want to find the angle at which the height is 44 meters. Rearranging the formula, we get: cos(theta) = (r - h) / r.

4. Plugging in the values, we have: cos(theta) = (35 - 44) / 35 = -9 / 35.

5. To find the angle theta, we take the inverse cosine (cos^-1) of this value: theta = cos^-1(-9 / 35) ≈ 102.88 degrees.

6. Now we need to determine the time it takes for the Ferris wheel to rotate through this angle. Since the Ferris wheel completes one rotation in 5 minutes, the time taken for 360 degrees is 5 minutes.

7. We can set up a proportion to find the time taken for 102.88 degrees: 102.88 degrees / 360 degrees = t / 5 minutes, where t is the time in minutes.

8. Solving for t, we get: t = (102.88 degrees / 360 degrees) * 5 minutes ≈ 1.45 minutes (rounded to two decimal places).

Therefore, approximately 1.45 minutes of the ride are spent higher than 44 meters above the ground.

To find out how many minutes of the ride are spent higher than 44 meters above the ground, we need to determine the height of the Ferris wheel at any given time during its rotation.

The Ferris wheel has a diameter of 70 meters, which means its radius is half of that, or 35 meters. This means the center of the Ferris wheel is 35 meters above the ground.

Since the boarding platform is 1 meter above the ground, the lowest point on the Ferris wheel is 1 meter + 35 meters = 36 meters above the ground.

As the Ferris wheel rotates, it completes one full revolution every 5 minutes. This means that every 5 minutes, we return to the same position.

To find out how much time is spent higher than 44 meters above the ground, we need to calculate the portion of the circle's circumference that goes above this height.

The height of 44 meters above the ground is 44 - 36 = 8 meters above the lowest point on the Ferris wheel.

The circumference of a circle is calculated using the formula: C = 2 * π * r, where C is the circumference and r is the radius.

Plugging in the values, we have C = 2 * 3.14 * 35 = 219.8 meters.

To find the fraction of the total circumference that is above 8 meters, we divide the height of 8 meters by the circumference: 8 / 219.8 ≈ 0.036.

This means that approximately 0.036 or 3.6% of the total ride time is spent higher than 44 meters above the ground.

To determine the number of minutes spent, we multiply the total ride time by the fraction above 44 meters: 0.036 * 5 minutes = 0.18 minutes.

Therefore, approximately 0.18 minutes, or about 11 seconds, of the ride are spent higher than 44 meters above the ground.

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