A ferris wheel is 250 ft in diameter and revolves every 40 second when in motion. You step up to the seat on the wheel at the bottom 2 feet about the fround so you are sitting 4 feet about the ground at start. Derive the formula for the height of your seat at time (t).
Then ask:
If i go around three times around, how long is the ride in distance traveled?
2356.194
124sinÐ/20(t-10)+126 or
124cosÐ/20(t)+126
To derive the formula for the height of your seat at time (t), we can use the equation of a circle to represent the motion of the ferris wheel. Let's consider the bottommost point of the wheel as the origin (0, 0) on a coordinate plane.
The radius of the ferris wheel is half of its diameter, which is 250 ft / 2 = 125 ft. Since you step up to the seat at 4 feet above the ground, your position can be represented as (0, 125 + 4) = (0, 129).
Now, the equation of a circle with center (h, k) and radius r is:
(x - h)^2 + (y - k)^2 = r^2
Plugging in the values we know, the equation becomes:
(x - 0)^2 + (y - 129)^2 = 125^2
Simplifying, we have:
x^2 + (y - 129)^2 = 15625
Since the ferris wheel revolves every 40 seconds, we can express x and y in terms of time t. Let's assume that at t = 0, your seat is at the bottommost position. At this point, the angle between the radius and the horizontal axis is 0. As time progresses, the angle increases, and we can express x and y as:
x = 125 * cos(2πt / 40)
y = 125 * sin(2πt / 40) + 129
Substituting these values into the equation of the circle, we get:
(125 * cos(2πt / 40))^2 + (125 * sin(2πt / 40) + 129 - 129)^2 = 15625
Simplifying further:
15,625 * cos^2(2πt / 40) + 15,625 * sin^2(2πt / 40) = 15625
Since cos^2(θ) + sin^2(θ) = 1, we have:
15,625 * (cos^2(2πt / 40) + sin^2(2πt / 40)) = 15625
15,625 = 15625
Therefore, the expression is true for all values of t.
Now, for the second part of your question: If you go around three times on the ferris wheel, you would complete three full revolutions, which is 3 * 2π radians. Since the radius of the ferris wheel is 125 ft, the distance you would travel is given by the formula:
Distance = 3 * 2π * 125 = 6π * 125 ≈ 2356.194 ft
To derive the formula for the height of your seat at a given time (t) on the ferris wheel, we can start by analyzing some key information:
1. The diameter of the ferris wheel is 250 ft, which means the radius (r) is half of the diameter, so r = 250 ft / 2 = 125 ft.
2. The ferris wheel completes one full revolution every 40 seconds, which means it has a period (T) of 40 seconds.
3. You start at a height of 4 ft above the ground when the ferris wheel is at the bottommost point, 2 ft above the ground.
To determine the height of your seat at a given time (t), we can use the equation of a sinusoidal function, specifically the sine function, because it completes one full period for T = 40 seconds.
The general equation for a sinusoidal function is:
y = A * sin(B * (x - C)) + D
where:
- A is the amplitude (half the difference between the maximum and minimum values of the function)
- B is the frequency (2π divided by the period T)
- C is the horizontal shift (time value when the function starts)
- D is the vertical shift (initial position)
In this case, the amplitude (A) is equal to the radius, r = 125 ft.
The frequency (B) can be calculated by dividing 2π by the period (T), so B = 2π / 40 seconds.
The horizontal shift (C) is 0 since we start at time t = 0 seconds.
The vertical shift (D) is the initial position, which is 4 ft above the ground.
Now we can write the formula for the height of your seat at time (t):
y = 125 * sin((2π / 40) * t) + 4
To find the distance traveled if you go around three times, we need to consider that each full revolution covers the circumference of the ferris wheel.
The circumference of a circle can be calculated using the formula C = 2πr, where r is the radius.
In this case, the radius (r) is 125 ft, so the distance traveled in one revolution is D1 = 2π * 125 ft.
If you go around three times, the total distance traveled is:
D_total = 3 * D1 = 3 * 2π * 125 ft ≈ 2356.194 ft.