The original ferris wheel, built by George Ferris for the Columbian Exposition of 1893, was much larger and slower than its modern counterparts: It has a diameter of 250 feet and contained 36 cars, each of which held 40 people; it made one revolution every 10 minutes. Suppose that the figure had coordinates (125,0) at time t=0. Find the rule of a function that gives the y- coordinate of car D at time t.

Question

The original ferris wheel, built by George Ferris for the Columbian Exposition of 1893, was much larger and slower than its modern counterparts: It has a diameter of 250 feet and contained 36 cars, each of which held 40 people; it made one revolution every 10 minutes. Suppose that the figure had coordinates (125,0) at time t=0. Find the rule of a function that gives the y- coordinate of car D at time t.

Answer 1

You need to tell me where D was at the start to get phi in

y = 125 + 125 sin (.2 pi t - phi)

Answer 2

125sin(pi(t)/5)

Answer 3

To find the rule of a function that gives the y-coordinate of car D at a given time t, we can make use of the information provided.

Let's break down the problem:

1. The original Ferris wheel had a diameter of 250 feet, which means its radius is 125 feet (since the diameter is twice the radius).
2. The Ferris wheel made one revolution every 10 minutes, which means it completed a full circle in 10 minutes.
3. At the start, when t = 0, car D is at coordinates (125, 0).

To find the rule of the function, we can assume that the Ferris wheel rotates counterclockwise. As time progresses, the Ferris wheel completes rotations. Each rotation completes in 10 minutes. In each rotation, every point on the Ferris wheel moves the same distance, forming a circle with center (0, 0) and radius 125.

Knowing this, we can establish the equation of a circle centered at the origin:

x^2 + y^2 = r^2,

where x and y are the coordinates at any given time t, and r is the radius of the Ferris wheel.

Now, let's substitute the given values:

x^2 + y^2 = 125^2.

Since the initial position of car D is (125, 0) at t = 0, we substitute these coordinates into the equation:

125^2 + 0^2 = 125^2.

Simplifying the equation gives us:

15625 = 15625.

This shows that the equation holds true for t = 0.

To find the rule of the function, we need to solve for y in terms of t. Rearranging the equation:

x^2 + y^2 = 125^2,

we get:

y^2 = 125^2 - x^2,

y = ±√(125^2 - x^2).

The negative square root is ignored since we are looking for the y-coordinate, and the value of y is always positive since the Ferris wheel is above the ground.

Therefore, the rule of the function that gives the y-coordinate of car D at time t is:

y = √(125^2 - x^2).