A Ferris wheel is 25 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 6 minutes. How many minutes of the ride are spent higher than 21 meters above the ground?
help me
the height is at the bottom at the start, so, since cos(x) is a maximum at x=0, you want something of the form
y = a - b*cos(kx)
25 meters in diameter, so
y = a - 25/2 cos(kx)
platform that is 4 meters above the ground
y = (4 + 25/2) - 25/2 cos(kx)
1 full revolution in 6 minutes, so 2π/k = 6
y = 33/2 - 25/2 cos(π/3 x)
now, when does the height = 21?
when x = 1.85 and 4.43
so it is above 21 for 4.43-1.85 = 2.58 minutes each cycle
To solve this problem, we need to find the vertical position of the Ferris wheel at any given time.
Step 1: Find the radius of the Ferris wheel.
The diameter is given as 25 meters, so the radius is half of that, which is 25/2 = 12.5 meters.
Step 2: Determine the height of the Ferris wheel at the 6 o'clock position.
The Ferris wheel is boarded from a platform that is 4 meters above the ground. At the 6 o'clock position, the bottom of the Ferris wheel is at ground level, so the height at that position is 4 meters.
Step 3: Calculate the distance traveled by the Ferris wheel in 1 minute.
The Ferris wheel completes 1 full revolution in 6 minutes. Therefore, it completes 1/6 of a revolution in 1 minute. The circumference of the Ferris wheel is 2πr, where r is the radius. Therefore, the distance traveled in 1 minute is (1/6)(2π(12.5)) = 13.09 meters.
Step 4: Determine the time spent higher than 21 meters.
To find the time spent higher than 21 meters, we can subtract the time spent below 21 meters from the total ride time.
To find the time spent below 21 meters, we need to determine the positions where the height is less than or equal to 21 meters.
The highest point on the Ferris wheel is when it is at the 12 o'clock position, where the height is the sum of the radius and the platform height: 12.5 + 4 = 16.5 meters.
The lowest point on the Ferris wheel is when it is at the 6 o'clock position, where the height is the platform height: 4 meters.
So, the Ferris wheel goes from a height of 16.5 meters to 4 meters in one revolution.
The Ferris wheel covers a distance of 4 + 16.5 = 20.5 meters in one revolution going from the highest to the lowest point.
The time spent below 21 meters is found by dividing this distance by the distance traveled in 1 minute: 20.5 / 13.09 ≈ 1.57 minutes.
Therefore, the time spent higher than 21 meters is the total ride time (6 minutes) minus the time spent below 21 meters (1.57 minutes):
6 - 1.57 ≈ 4.43 minutes.
So, approximately 4.43 minutes of the ride are spent higher than 21 meters above the ground.
To solve this problem, we need to figure out at which times during the ride the Ferris wheel is higher than 21 meters above the ground.
Let's start by finding the maximum height of the Ferris wheel. The diameter of the Ferris wheel is given as 25 meters, so the radius is half of that, which is 12.5 meters.
Since the loading platform is 4 meters above the ground, the total height from the ground to the highest point on the Ferris wheel is 12.5 meters (radius) + 4 meters (platform height) = 16.5 meters.
Next, we need to determine the lowest point on the Ferris wheel. Since the six o'clock position is level with the loading platform, the lowest point is when the Ferris wheel is half a revolution away from the six o'clock position. So the lowest point is 180 degrees away from the six o'clock position.
Now let's calculate how many minutes of the ride are spent higher than 21 meters above the ground. To do this, we need to find out the times when the Ferris wheel is at a height greater than 21 meters.
Since the wheel completes 1 full revolution in 6 minutes, we can divide the 360 degrees of a full revolution by 6 to find out how many degrees it moves per minute. 360 degrees / 6 minutes = 60 degrees per minute.
To find the times when the Ferris wheel is at a height greater than 21 meters, we need to determine the angles at the highest and lowest points. The highest point is at the 12 o'clock position (0 degrees) and the lowest point is at the six o'clock position (180 degrees).
Now we can calculate the angles at which the Ferris wheel is higher than 21 meters. To do this, we take the arcsin of (21/16.5) using a calculator. The arcsin of a value gives us the angle in radians, so we need to convert it to degrees.
Let's calculate the angles:
angle1 = arcsin(21/16.5) * (180/π)
angle2 = 180 - angle1
Now that we have the angles, we can calculate the time spent at a height greater than 21 meters. The Ferris wheel moves 60 degrees per minute, so we divide the angles by 60 to get the time in minutes.
time1 = angle1 / 60
time2 = angle2 / 60
To find the total time spent higher than 21 meters, we can add time1 and time2 together.
total_time = time1 + time2
And that's how you can calculate the number of minutes of the ride spent higher than 21 meters above the ground.