A Ferris wheel is 32 meters in diameter and makes one revolution every 4 minutes. For how many minutes of any revolution will your seat be above 24 meters?
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r = 16 , I assume grazes ground
h = 16 - 16 cos T where T is angle up from straight down
h = 16(1 - cos T)
let's find out what angle hits 24 meters
24 = 16 (1 - cos T)
1.5 - 1 = - cos T
cos T = -.5
T = 120 degrees
so between 120 and (360 - 120) = 240
which is 120 or 1/3 of the way around
4 minutes * (1/3 of revolution ) = 1 1/3 minutes
Well, let's do some math here. The diameter of the Ferris wheel is 32 meters, so the radius would be half of that, which is 16 meters.
Since the Ferris wheel makes one revolution every 4 minutes, we can calculate the circumference of the wheel, which is 2πr. Plugging in the radius, we get 2 x 3.14 x 16 = 100.48 meters.
Now, we need to figure out for how long the seat will be above 24 meters.
Since the wheel completes one full revolution in 4 minutes, we can divide that time by the circumference to figure out how long it takes to travel one meter.
So, we have 4 minutes / 100.48 meters = 0.04 minutes per meter.
Now, if we want to find out how long the seat will be above 24 meters, we can multiply the time per meter by 24.
0.04 minutes per meter x 24 meters = 0.96 minutes.
Therefore, your seat will be above 24 meters for approximately 0.96 minutes during any revolution of the Ferris wheel. Enjoy the view!
To determine the time your seat will be above 24 meters, we can calculate the angle corresponding to this height.
1. Find the radius of the Ferris wheel: The diameter is given as 32 meters, so the radius will be half of that, which is 32/2 = 16 meters.
2. Determine the circumference of the Ferris wheel: The circumference of a circle is given by the formula C = 2πr, where r is the radius. Let's calculate it: C = 2π(16) = 32π meters.
3. Calculate the fraction of the circle representing 24 meters: To find the angle, we need to determine what fraction of the circumference corresponds to 24 meters. This can be calculated using the formula: fraction = height / circumference = 24 / (32π) = 3 / (4π).
4. Convert the fraction into degrees: To find the angle in degrees, we multiply the fraction by 360° since a complete circle is 360°. The formula is: angle in degrees = fraction * 360° = (3 / (4π)) * 360°.
5. Evaluate the expression: Let's calculate the angle in degrees.
angle in degrees = (3 / (4π)) * 360°
angle in degrees ≈ (3 / (4*3.14)) * 360°
angle in degrees ≈ (3 / 12.56) * 360°
angle in degrees ≈ (0.2389) * 360°
angle in degrees ≈ 86.00°
Therefore, your seat will be above 24 meters for approximately 86 degrees of any revolution, which can also be expressed as approximately 4.77 minutes (since the wheel takes 4 minutes to complete one revolution).
To find out for how many minutes your seat will be above 24 meters during a revolution of the Ferris wheel, we first need to understand the height of the seat at different positions during one revolution.
The Ferris wheel has a diameter of 32 meters, which means it has a radius of half that, or 16 meters. The height of the seat can be calculated using the equation of a circle, where the height is the square root of the radius squared minus the distance from the center squared.
So, the equation is: height = sqrt(radius^2 - distance^2)
In this case, the radius is 16 meters. We need to find out the distance from the center of the Ferris wheel when the seat is at a height of 24 meters.
Using the equation above:
24 = sqrt(16^2 - distance^2)
Solving for distance:
576 = 256 - distance^2
distance^2 = 256 - 576
distance^2 = -320
Since we cannot have a negative distance, it means that the seat does not reach a height of 24 meters during one revolution of the Ferris wheel.
Therefore, for all 4 minutes of a revolution, your seat will not be above 24 meters.