Problem 54 - Algebra A Combined Function

Question

Although there are many larger observation wheels on the horizon, as of this writing the largest observation wheel in the world is the Singapore Flyer. From the Flyer, you can see up to \(45 \mathrm{~km}\) away. Each of the 28 enclosed capsules holds 28 passengers, completes a full rotation every 32 minutes. Its diameter is 150 meters, and the height of this giant wheel is 165 meters. (Source: singaporeflyer.com) a. What is the radius of the Singapore Flyer? b. How close is the wheel to the ground? C. How high is the center of the wheel from the ground? d. Using the axes in the drawing, what are the coordinates of the center of the wheel? e. Use parts a and \(\mathbf{d}\) to write an equation of the Singapore Flyer.

Step-by-Step Solution

Answer

a. 75 m b. 90 m c. 127.5 m d. (0, 127.5) e. \(x^2 + (y - 127.5)^2 = 5625\)

Step 1: Calculate the Radius

To find the radius, we need to take half of the diameter of the wheel. The diameter is given as 150 meters, so the radius \( r \) is \( \frac{150}{2} = 75 \) meters.

Step 2: Distance from Ground to Bottom of Wheel

To find how close the bottom of the wheel is to the ground, subtract the radius from the total height of the wheel. The height is 165 meters, so the distance is \( 165 - 75 = 90 \) meters.

Step 3: Calculate Height of the Center

Since the distance from the bottom of the wheel to the top is twice the radius, the center of the wheel is the radius above the bottom. Therefore, the center's height is \( 165 - 75/2 \), which equals \( 90 + 75/2 = 127.5 \) meters.

Step 4: Determine Coordinates of the Center

Assuming the wheel is centered on a horizontal axis and rests on the ground, the coordinates of the center can be given as \((0, \text{height from ground})\). Therefore, the coordinates are \((0, 127.5)\) meters.

Step 5: Write the Equation of the Circle

The general form of a circle's equation is \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius. Substituting for the Singapore Flyer, we have \((x - 0)^2 + (y - 127.5)^2 = 75^2\). Simplifying gives \(x^2 + (y - 127.5)^2 = 5625\).

Key Concepts

Radius

The radius is an important measure in circular motion as it helps define the size of the circle. In the context of a real-world example like the Singapore Flyer, the radius is half of the diameter. This is because the diameter spans the entire width of the circle through its center, while the radius only stretches from the center to the edge. In mathematical terms, the radius (\(r\)) can be calculated using the formula: \[ r = \frac{d}{2} \] where \(d\) is the diameter. For the Singapore Flyer, with a diameter of 150 meters, the radius is: \[ r = \frac{150}{2} = 75 \text{ meters} \] Knowing the radius helps in many calculations related to circular motion, like finding the circumference or the area of the wheel. It also feeds into other calculations related to the wheel's position relative to the ground.

Equation of a Circle

The equation of a circle in a Cartesian coordinate system embodies its geometric properties and helps determine the position and path of circular motion. The standard equation of a circle is given as: \[ (x - h)^2 + (y - k)^2 = r^2 \] Here, \((h, k)\) represents the circle's center, and \(r\) is the radius. This equation essentially describes all the points \((x, y)\) that make up the circle based on their distance to the center, which will always equal the radius. For the Singapore Flyer, where - the center is positioned at \((0, 127.5)\)- the radius is 75 meters, the equation becomes: \[ (x - 0)^2 + (y - 127.5)^2 = 75^2 \] Simplifying this gives: \[ x^2 + (y - 127.5)^2 = 5625 \] This allows you to understand and visualize the motion and structure of the Flyer as a part of its underlying geometry.

Distance from Ground

The distance from the ground to different parts of a large structure like the Singapore Flyer is crucial for understanding its engineering and safety aspects. To determine how close the bottom of the wheel is to the ground, you would subtract the radius from the entire height of the wheel above ground level. For the Singapore Flyer, with a height of 165 meters and a radius of 75 meters, this distance is calculated as: \[ 165 - 75 = 90 \text{ meters} \] This calculation ensures that we understand how elevated the structure's rotation path is from the ground, affecting both engineering decisions and visitor safety. The height at which the wheel's center lies can also be calculated using similar principles. Once the bottom is 90 meters from the ground and the radius is 75 meters above that point, the center is precisely halfway along the circle, ensuring an equidistant circular path around the central axis.