A Ferris wheel is 38 meters in diameter and the bottom of the Ferris wheel is 6 meters above the ground. You board the Ferris wheel at the 3 o'clock position.
The wheel completes one full revolution every 2.2 minutes. What is the angular speed (in radians per minute) that the Ferris wheel is rotating?
2.855993
Correct radians per minute
Write a formula that gives the angle measure (in radians) swept out from the 3 o'clock position, a, in terms of the number of minutes elapsed since you boarded the Ferris wheel, t
.
Define a function f that gives your height above the ground (in meters) in terms of the number of minutes elapsed since you boarded the Ferris wheel,
t
.
so the axle is at 38/2 + 6 = 25m up.
θ covers 2π radians every 2.2 minutes, so the angular speed is
ω = 2πrad/2.2min = π/1.1 rad/min
thus θ = π/1.1 t
Since you are rising from there, you have
h = 25 + 19sin(ωt) = 25 + 19sin(π/1.1 t)
Sure, let's go ahead and define the formulas.
The angular speed (in radians per minute) can be calculated using the formula:
Angular speed = (2π radians) / (time taken for one revolution in minutes)
In this case, the time taken for one revolution is 2.2 minutes. So, plugging in the values:
Angular speed = (2π radians) / (2.2 minutes) ≈ 2.855993 radians per minute
For the angle measure swept out from the 3 o'clock position, we can use the formula:
Angle measure = (Angular speed) * (time elapsed since boarding)
So, the formula is:
a = (2.855993 radians per minute) * t
Now, for your height above the ground, we can use the sine function to calculate it. The formula is:
f(t) = (radius of the Ferris wheel) * sin(a)
Since the Ferris wheel has a diameter of 38 meters, the radius is half of that, which is 19 meters. The angle measure, a, can be calculated using the formula above.
So, the final formula is:
f(t) = 19 * sin[(2.855993 radians per minute) * t]
To find the angular speed of the Ferris wheel, we need to determine the angle swept out by the wheel in one minute.
Given that the Ferris wheel completes one full revolution (360 degrees or 2π radians) every 2.2 minutes, we can calculate the angular speed as follows:
Angular speed = (2π radians) / (2.2 minutes)
Angular speed ≈ 2.855993 radians per minute
Therefore, the angular speed of the Ferris wheel is approximately 2.855993 radians per minute.
To define a formula for the angle measure swept out from the 3 o'clock position, a, in terms of the number of minutes elapsed since you boarded the Ferris wheel, t, we can use the fact that the Ferris wheel completes one full revolution every 2.2 minutes.
Since there are 2π radians in one revolution, we can define:
a = (2π radians / 2.2 minutes) * t
This formula gives the angle measure, a, in radians, based on the number of minutes, t, that have elapsed since you boarded the Ferris wheel.
To define a function f that gives your height above the ground in terms of the number of minutes elapsed, t, we can use the trigonometric relationship between the angle and the height of the Ferris wheel.
Let h be the height above the ground.
Since the Ferris wheel has a diameter of 38 meters and the bottom is 6 meters above the ground, the radius is (38 meters / 2) - 6 meters = 13 meters.
Using the radius and the angle measure, we can define the function as follows:
f(t) = 13 * sin(a)
Thus, the function f gives your height above the ground, in meters, based on the number of minutes, t, that have elapsed since you boarded the Ferris wheel.
To solve the problem, we can first calculate the circumference of the Ferris wheel, which is equal to π times the diameter. In this case, the diameter is 38 meters, so the circumference is 38π meters.
Since the Ferris wheel completes one full revolution every 2.2 minutes, we can calculate the angular speed by dividing the circumference by the time taken for one revolution. The formula for angular speed (ω) is:
ω = (2π) / (time taken for one revolution)
Substituting the values, we have:
ω = (2π) / 2.2
Evaluating this expression, we get:
ω ≈ 2.855993 radians per minute
So, the angular speed of the Ferris wheel is approximately 2.855993 radians per minute.
Now, to define a formula for the angle measure (a) swept out from the 3 o'clock position in terms of the number of minutes elapsed (t), we can use the formula:
a = (ω * t)
Substituting the previously calculated value of ω, we have:
a = (2.855993 * t)
This formula gives us the angle measure (in radians) swept out from the 3 o'clock position, a, in terms of the number of minutes elapsed since you boarded the Ferris wheel, t.
To define a function f that gives your height above the ground (h) in terms of the number of minutes elapsed since you boarded the Ferris wheel (t), we can exploit the relationship between angle measure and height.
Since the Ferris wheel is a circle, and you start at the 3 o'clock position, the angle (a) swept out from the 3 o'clock position is directly related to your height above the ground (h). We can use trigonometry to determine this relationship.
Let's consider a right triangle formed by your height above the ground (h), the radius of the Ferris wheel (r = 19 meters), and the angle (a) swept out from the 3 o'clock position. The height (h) is the opposite side, and the radius (r) is the adjacent side.
Using the trigonometric function tangent (tan), we have:
tan(a) = h/r
Rearranging the equation to solve for h, we get:
h = r * tan(a)
Substituting the formula for a as (2.855993 * t) and the radius (r = 19 meters), we have:
h = 19 * tan(2.855993 * t)
This formula gives your height above the ground (h) in meters in terms of the number of minutes elapsed since you boarded the Ferris wheel (t).
So, the formula f(t) = 19 * tan(2.855993 * t) represents the function that gives your height above the ground in meters as a function of time in minutes since you boarded the Ferris wheel.