A ferris wheel is 30 meters in diameter and boarded in the six o'clock position from a platform that is 10 meters above the ground. The wheel completes one full revolution every 10 minutes. At the initial time t=0 you are in the twelve o'clock position.Find a formula, using the cosine function, for your height above ground after t minutes on the ferris wheel.
Find two numbers the exact answer is between 6x7,381
To find a formula for your height above the ground after t minutes on the ferris wheel, we can start by determining the equation for the height of a point on the wheel as it rotates.
Let's consider the center of the ferris wheel as the origin of a coordinate system, with the x-axis and the y-axis aligned as follows: the positive x-axis points to the three o'clock position, and the positive y-axis points to the twelve o'clock position.
Since the ferris wheel completes one full revolution every 10 minutes, its angular speed can be calculated as (2π radians) / (10 minutes).
To find your height above the ground after t minutes, we need to use the cosine function to determine the y-coordinate of your position on the ferris wheel. The cosine function relates the angle θ (in radians) to the x-coordinate of a point on the unit circle.
In this case, we need to determine the angle θ based on the current time t.
As the ferris wheel completes one full revolution every 10 minutes, the angle θ (in radians) can be calculated as:
θ = (2π / 10) * t
To account for the starting position at the twelve o'clock position, the initial angle should be set to π / 2 (90 degrees).
The formula for the height above the ground after t minutes on the ferris wheel can be expressed as:
height(t) = 10 + (distance from center to point on wheel) * cos(angle)
The distance from the center to a point on the wheel is half the diameter, which is 30 / 2 = 15 meters.
Plugging in the values, the formula becomes:
height(t) = 10 + 15 * cos(θ)
Substituting the value for θ, we have:
height(t) = 10 + 15 * cos((2π / 10) * t + π / 2)
Therefore, the formula for your height above the ground after t minutes on the ferris wheel using the cosine function is height(t) = 10 + 15 * cos((2π / 10) * t + π / 2).
To find a formula for your height above ground after t minutes on the ferris wheel, we will first need to analyze the situation.
The diameter of the ferris wheel is given as 30 meters, which means its radius is half of that, or 15 meters. The ferris wheel completes one full revolution every 10 minutes.
At t=0, you are in the twelve o'clock position, which is also known as the zero degree position. As time progresses, the ferris wheel rotates counterclockwise.
To use the cosine function to find your height above the ground, we can consider the position of the ferris wheel in terms of angles. Since the ferris wheel completes one full revolution every 10 minutes, the angular velocity (ω) of the ferris wheel can be calculated as:
ω = (2π radians) / (10 minutes)
Let's assume that your height above the ground at the twelve o'clock position is h = 10 meters (since the platform is 10 meters above the ground).
Now, let's find an expression for your height above the ground, h(t), after t minutes using the cosine function.
The general form of the cosine function is given by:
cos(θ) = adjacent / hypotenuse
or
cos(θ) = x / r
In this case, we want to find the height above ground, h(t), which corresponds to the adjacent side of a right triangle formed by the radius and the height at any given angle θ.
Since the ferris wheel is rotating counterclockwise, the initial position at t = 0 corresponds to θ = 0 degrees, which is equivalent to π/2 radians (since the vertical position is cos(π/2) = 0).
Therefore, the angle of the ferris wheel at any given time t can be calculated as:
θ = (ω * t) + (π/2)
Now, substituting the values into the cosine function:
cos(θ) = h(t) / r
cos[(ω * t) + (π/2)] = h(t) / r
Rearranging the equation, we get:
h(t) = r * cos[(ω * t) + (π/2)]
Substituting the values of r, ω, and rearranging for the final formula, we have:
h(t) = 15 * cos[(2π/10) * t + (π/2)]
Therefore, the formula, using the cosine function, for your height above ground after t minutes on the ferris wheel is h(t) = 15 * cos[(π/5) * t + (π/2)].
This formula will give you the height above ground (h) at any given time (t) on the ferris wheel.
at the top when t=0, so no phase shift
h = Acos(at)+k
center of wheel is at 10+30/2 = 25, so
h = Acos(at)+25
wheel radius is 15, so
h = 15cos(at)+25
period = 10, so use 2π/10
h = 15cos(πt/5)+25