 A passenger must climb 7 steps to a platform to enter the gondola. Each step is 7.5 inches. What
would the equation of the ferris wheel circle be on a cartesian coordinate graph if the x-axis is the
ground and the y-axis is the axis of symmetry of the ferris wheel.


To find the equation of the Ferris wheel circle on a Cartesian coordinate system, we need the following information:

1. The **center of the Ferris wheel** (the coordinates of the center).
2. The **radius** of the Ferris wheel.

### Step 1: Height of the platform
The height of the platform can be calculated by multiplying the number of steps by the height of each step. There are 7 steps, each 7.5 inches high:

\[
\text{Height of platform} = 7 \times 7.5 = 52.5 \, \text{inches} = \frac{52.5}{12} \, \text{feet} \approx 4.375 \, \text{feet}
\]

This height is added to the Ferris wheel's ground clearance to determine where the passenger boards the gondola.

### Step 2: Setting up the equation of the circle
Assuming that the Ferris wheel is perfectly vertical, the center of the circle will be on the y-axis. The equation of a circle on a Cartesian graph is given by:

\[
(x - h)^2 + (y - k)^2 = r^2
\]

Where:
- \((h, k)\) is the center of the circle.
- \(r\) is the radius of the Ferris wheel.

#### Assumptions:
- If the center of the Ferris wheel is at height \(k\) feet above the ground, it will include both the height of the platform and half the Ferris wheel's diameter (its radius).
- The axis of symmetry is along the y-axis, so the x-coordinate of the center is 0 (i.e., \(h = 0\)).

Let’s say the **radius of the Ferris wheel** is \(r\) feet (you will need to provide this for an exact equation). The height of the center of the Ferris wheel would be:

\[
k = \text{Height of platform} + r = 4.375 + r \, \text{feet}
\]

### Step 3: Final equation
Thus, the equation of the Ferris wheel will be:

\[
x^2 + (y - (4.375 + r))^2 = r^2
\]

Where \(r\) is the radius of the Ferris wheel.

### Example
If the Ferris wheel has a radius of, say, 50 feet, then the equation would be:

\[
x^2 + (y - (4.375 + 50))^2 = 50^2
\]
\[
x^2 + (y - 54.375)^2 = 2500
\]

This is the equation of the Ferris wheel circle.

---

Do you want more details on how to find the radius or solve a specific case?

**Related Questions:**
1. What is the Ferris wheel's full height if we know its diameter?
2. How would the equation change if the Ferris wheel is tilted?
3. How does changing the step height affect the platform height and equation?
4. Can we model the position of the gondola as it rotates around the circle?
5. How can we calculate the angular velocity of the Ferris wheel?

**Tip:** The general form of a circle equation is highly versatile for modeling real-world scenarios like Ferris wheels! Always focus on the coordinates of the center and the radius.

A passenger must climb 7 steps to a platform to enter the gondola. Each step is 7.5 inches. What would the equation of the ferris wheel circle be on a cartesian coordinate graph if the x-axis is the ground and the y-axis is the axis of symmetry of the ferris wheel?

Learn how to calculate the equation of a Ferris wheel's circle on a Cartesian coordinate graph. This involves using geometry and algebra to model the Ferris wheel, incorporating the platform height and the radius of the wheel. The problem covers circle equations and coordinate systems, suitable for students in Grades 9-12.

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