A Ferris wheel is 25 meters in diameter and boarded from a platform that is 2 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 4 minutes. How many minutes of the ride are spent higher than 21 meters above the ground?
To solve this problem, we need to find the height of the Ferris wheel at the 6 o'clock position, and then determine how much time it spends higher than 21 meters above the ground during one revolution.
1. Find the height of the Ferris wheel at the 6 o'clock position:
- The diameter of the Ferris wheel is 25 meters, so the radius is half of that, which is 25/2 = 12.5 meters.
- The loading platform is 2 meters above the ground, so at the 6 o'clock position, the height of the Ferris wheel is 12.5 + 2 = 14.5 meters.
2. Calculate the circumference of the Ferris wheel:
- The circumference of a circle can be found using the formula C = 2πr, where r is the radius.
- The radius of the Ferris wheel is 12.5 meters, so the circumference is 2π * 12.5 = 25π meters.
3. Calculate the time it takes to complete one revolution:
- The Ferris wheel completes one full revolution in 4 minutes.
4. Determine the height above 21 meters during one revolution:
- The highest point of the Ferris wheel is when it is at the 12 o'clock position.
- So, the Ferris wheel spends half of its time above 21 meters during one revolution.
- The height of the 12 o'clock position is equal to the radius plus the loading platform height, which is 12.5 + 2 = 14.5 meters.
- Therefore, the Ferris wheel spends half of 4 minutes, which is 2 minutes, above 21 meters during one revolution.
Therefore, the Ferris wheel spends 2 minutes of the ride higher than 21 meters above the ground.
To solve this problem, we will first need to determine the height of the Ferris wheel for any given position. Then, we can calculate the time spent higher than 21 meters above the ground.
Let's start by finding the height of the Ferris wheel for the highest point it reaches. The diameter of the Ferris wheel is 25 meters, so the radius (r) can be calculated by dividing the diameter by 2:
r = 25 / 2 = 12.5 meters
Since the loading platform is 2 meters above the ground, the total height of the Ferris wheel from the ground to its highest point is r + 2:
Total height = 12.5 + 2 = 14.5 meters
Now, we need to find the difference between the total height of the Ferris wheel and the height of 21 meters, as we are interested in the time spent higher than 21 meters. Let's call this difference D:
D = Total height - 21 = 14.5 - 21 = -6.5 meters
Since the difference is negative, it means that the Ferris wheel never reaches a height of 21 meters or higher. Therefore, the answer is 0 minutes.
In summary, none of the ride time is spent higher than 21 meters above the ground.
Draw Cartesian system.
Height from the origin of Cartesian system to center of a circle = height of a platform + radius of the wheel
h = 2 + 12.5
h = 14.5 m
At point x = 0 , y = 14.5 draw a circle whose radius is:
r = 25 / 2 = 12.5 m
If total height of the cabin > 21 m then height above horisontal axis of a circle must be:
h > 21 - 14.5
h > 6.5 m
Angular speed:
ω = angle / tme
ω = 360° / 4 min
ω = 90° / min
t1 = time the wheel cabin takes to reach the starting position from 90°:
t1 = 90° / ω = 90° / ( 90° / min ) = 1 min
Mark the angle between the horizontal axis of a circle and the wheel cabin with θ.
Now:
sin θ = y / r
sin θ = y / 12.5
y = 12.5 ∙ sin θ
12.5 ∙ sin θ = 6.5
sin θ = 6.5 / 12.5
sin θ = 0.52
t2 = time spent by the cabin to reach the position h = 6.5 m
ω = angle / tme
ω = θ / t2
θ = t2 ∙ ω
Now you must solve:
12.5 ∙ sin θ = 6.5
12.5 ∙ sin ( t2 ∙ ω ) = 6.5
sin ( t2 ∙ 90° / min ) = 6.5 / 12.5 = 0.52
( t2 ∙ 90° / min ) = sin⁻¹ ( 0.52 )
t2 ∙ 90° / min = 31.3322515°
t2 = 31.3322515° / ( 90° / min ) = 0.348136127751 min
Total time for height > 21 m
t > t1 + t2
t > 1 + 0.348136127751
t > 1.348136127751min
P.S.
Sorry for my bad English.
aim for an equation of the type
height = a sin k(t + d) + c
where the variables are probably defined in your text or your class notes.
25 meters in diameter ----> a = 12.5
minimum is 2 ----> c = 13.5
1 full revolution in 4 minutes ----> 2π/k = 4, ----> k = π/2
so lets start with
h = 12.5 sin π/2(t + d) + 13.5
when t = 0 , we want h = 2
12.5 sin π/2(0 + d) + 13.5 = 2
sin π/2(d) = -.92
using my calculator:
π/2(d) = -1.1681..
d = -.7436..
height = 12.5sin π/2(t - .7436) + 13.5
looks good:
https://www.wolframalpha.com/input/?i=y+%3D+12.5sin(%CF%80%2F2(x+-+.7436))+%2B+13.5
Now I leave it up to you to solve:
12.5sin π/2(t - .7436) + 13.5 > 21