Question

The High Roller observation wheel in Las Vegas has a maximum height of 550 feet and a diameter of 520 feet, with one full rotation taking approximately 30 minutes. Find an equation for the wheel if the center of the wheel is on the \(y\) -axis.

Short answer

The equation of the wheel is x^2 + (y - 275)^2 = 67600.

Step-by-step solution

Determine the Center of the Wheel

The center of the wheel lies on the y-axis at half the maximum height, which is 550 feet. The radius of the wheel is half the diameter (520 feet), which is 260 feet. So, the center is at : (0, 550/2) = (0, 275)

Write the Equation of the Circle

A circle with center (h, k) and radius r is represented as (x - h)^2 + (y - k)^2 = r^2. Here h = 0, k = 275, and r = 260. Thus, the equation becomes (x - 0)^2 + (y - 275)^2 = 260^2, simplifying to x^2 + (y - 275)^2 = 67600.

Key concepts

radius of a circle

The radius of a circle is the distance from the center of the circle to any point on its perimeter. In this exercise, the diameter of the High Roller observation wheel is provided as 520 feet. The radius is simply half of the diameter.
Therefore, we calculate the radius as follows: \[ \text{Radius} = \frac{\text{Diameter}}{2} = \frac{520 \text{ feet}}{2} = 260 \text{ feet} \]
Understanding the radius is crucial because it's a fundamental part of the circle's equation. The radius helps us identify how far out we need to go from the center to the edge of the circle.

coordinate geometry

Coordinate geometry, or analytic geometry, involves placing geometric figures on the coordinate plane and using algebra to solve geometric problems. In our exercise, we are dealing with a circle whose center is on the vertical (\text{y-axis}).
The center of our High Roller wheel lies halfway up its maximum height. Here's why:

• Maximum height: 550 feet
• Center is halfway, so: \[ \text{Center} = \frac{550 \text{ feet}}{2} = 275 \text{ feet} \]

We place the center at (0, 275) because the center lies on the y-axis. Hence, the coordinates (0, 275) represent the center of the High Roller observation wheel.

trigonometric applications in geometry

Trigonometry often comes into play with circles, especially when we need to find points around the circle or angles subtended by arcs. In our scenario, we can use the basic circle equation derived from geometry:
A circle with the center at \((h, k)\) and radius \(r\) is given by:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
In our given problem:

Substitute the Known Values

• Center: \( (0, 275) \)
• Radius: \( 260 \text{ feet} \)

Plug these values into our circle equation:
\[ (x - 0)^2 + (y - 275)^2 = 260^2 \]
Solve further to get: \[ x^2 + (y - 275)^2 = 67600 \]

This is our final equation for the High Roller wheel in Las Vegas.

Additionally, understanding trigonometric fundamentals can also help us calculate time slices, arc lengths, and positions on the circle as it rotates.