A Ferris wheel is 10 meters in diameter and boarded from a platform that is 3 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 11 meters above the ground?
make a sketch of the positions at t = 0, 1, 2, 3, and 4
and you will see that you need something like
height = 5sin(π/2)t + 8
to meet your condition of 6:00 o'clock position being 3 m
we do a 1 second shift to the right,
height = 5sin [(π/2)(t-1)] + 8
check: when t = 0 , height = 5sin( -π/2)+8 = 3
when t = 1, height = 8
when t = 2, height = 5sin π/2 + 8 = 13
..
when t=4, height = 5sin π + 8 = 3
equation is correct,
so we want
5sin [(π/2)(t-1)] + 8≥11
(lets use the =)
5sin [(π/2)(t-1)] + 8 = 11
5sin [(π/2)(t-1)] = 3
sin [(π/2)(t-1)] = .6
I know sin (.6435) = appr .6
π/2(t-1) = .6435 or π/2(t-1) = π-.6435
t-1 =.4096655.. or t-1 = 1.59033..
t = appr 1.41 or t = appr 2.59
so time spent higher than 11 m is
2.59-1.41 seconds or
appr 1.18 seconds
To solve this problem, we need to find the angular position of the Ferris wheel at a height above 11 meters.
1. First, let's find the radius of the Ferris wheel. The diameter is given as 10 meters, so the radius is half of that, which is 10/2 = 5 meters.
2. Next, let's find the circumference of the Ferris wheel. The circumference can be calculated by multiplying the diameter by π (pi). So, the circumference of the Ferris wheel is 10π meters.
3. Since the Ferris wheel completes 1 full revolution in 4 minutes, we can calculate the angular speed of the wheel. The angular speed can be calculated by dividing 360 degrees (a full revolution) by 4 minutes. So, the angular speed is 360/4 = 90 degrees per minute.
4. Now, let's find the angle between the six o'clock position and the position at a height above 11 meters. Since the wheel completes 1 full revolution in 4 minutes, the angle between the six o'clock position and any other position on the wheel is directly proportional to the time elapsed. Thus, we can set up a proportion:
(angle at a height above 11 meters) / (360 degrees) = (time spent above 11 meters) / (4 minutes)
Let's say the angle at a height above 11 meters is θ degrees.
θ / 360 = (time spent above 11 meters) / 4
5. We know that the height of the wheel at any given angle is given by the equation h = r sin(θ), where h is the height, r is the radius, and θ is the angle in radians. We can convert the height to meters by adding the height of the loading platform, which is 3 meters.
So, for a height above 11 meters, we have the equation:
11 = 5 sin(θ) + 3
6. Rearranging the equation, we have:
8 = 5 sin(θ)
sin(θ) = 8/5
7. Using the inverse sine function, we can find the value of θ. The inverse sine of 8/5 is approximately 0.927.
8. Now, substituting this value of θ in the proportion we set up earlier, we have:
0.927 / 360 = (time spent above 11 meters) / 4
9. Cross-multiplying, we get:
(time spent above 11 meters) = (0.927 / 360) * 4
10. Calculating the result, we have:
time spent above 11 meters = 0.0103 minutes
Therefore, approximately 0.0103 minutes (or about 0.62 seconds) of the ride are spent higher than 11 meters above the ground.
To find out how many minutes of the ride are spent higher than 11 meters above the ground, we need to analyze the height of the Ferris wheel at different positions.
First, let's determine the highest point of the Ferris wheel. The diameter of the Ferris wheel is 10 meters, so the radius is half of that, which is 5 meters.
When the wheel is at the twelve o'clock position, it reaches its highest point, which is 5 meters above the center of the wheel. Since the loading platform is 3 meters above the ground, the highest point of the Ferris wheel from the ground is 5 + 3 = 8 meters.
Next, we need to calculate how many degrees are covered in one minute of the ride. The wheel takes 4 minutes to complete one full revolution, which means it covers 360 degrees in 4 minutes. Thus, the Ferris wheel covers 360/4 = 90 degrees per minute.
Now, let's determine how many minutes of the ride are spent higher than 11 meters above the ground. To do this, we need to calculate the angle at which the 11-meter mark occurs.
We know that the highest point of the Ferris wheel is 8 meters above the ground. So, to reach 11 meters, we need to go 11 - 8 = 3 meters higher.
The radius of the Ferris wheel is 5 meters, and the height we want to reach is 3 meters higher than the highest point. Therefore, we can use simple trigonometry to find the angle corresponding to this rise in height.
sin(angle) = opposite/hypotenuse
sin(angle) = 3/5
angle = arcsin(3/5) ≈ 36.87 degrees
This means that at an angle of 36.87 degrees above the horizontal, the height of the Ferris wheel will be 11 meters. The wheel covers 90 degrees per minute, so the time taken to reach this angle can be calculated by dividing 36.87 by 90, and then multiplying by 1 minute:
time = (36.87/90) * 1 ≈ 0.4096 minutes
Therefore, approximately 0.41 minutes or 24.58 seconds of the ride are spent higher than 11 meters above the ground.