Problem 56 - Algebra A Combined Function

Question

The world's largest-diameter Ferris wheel currently operating is the Cosmo Clock 21 at Yokohama City, Japan. It has a 60 -armed wheel, its diameter is 100 meters, and it has a height of 105 meters. (Source: The Handy Science Answer Book) a. What is the radius of this Ferris wheel? 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 wheel.

Step-by-Step Solution

Answer

a. 50 m; b. 5 m; c. 55 m; d. (0, 55); e. \( x^2 + (y - 55)^2 = 2500 \).

Step 1: Find the radius of the Ferris wheel

The radius of a circle is half of its diameter. Given that the diameter of the Ferris wheel is 100 meters, the radius can be calculated as follows:\[ r = \frac{d}{2} = \frac{100}{2} = 50 \text{ meters} \]

Step 2: Calculate how close the wheel is to the ground

The wheel has a total height of 105 meters. To find out how close the bottom of the wheel is to the ground, subtract the radius from the total height:\[ \text{Distance to ground} = 105 - 100 = 5 \text{ meters} \]

Step 3: Determine the height of the center of the wheel above the ground

The center of the wheel is located at the center of its diameter. Since the diameter is 100 meters and the base is 5 meters from the ground, add the radius to the base's height:\[ \text{Center height} = 5 + 50 = 55 \text{ meters} \]

Step 4: Identify the coordinates of the wheel's center

Considering the wheel's center is vertically above the point where it is closest to the ground, its coordinates would be aligned with the height of the center and the horizontal position along the diameter. Assuming the horizontal position is centered at 0, the coordinates of the center are:\[ (0, 55) \]

Step 5: Write the equation of the wheel using the center and radius

The general equation for a circle is \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius. Using the center coordinates \((0, 55)\) and radius 50, the equation is:\[ (x - 0)^2 + (y - 55)^2 = 50^2 \]Simplifying, this becomes:\[ x^2 + (y - 55)^2 = 2500 \]

Key Concepts

Circle Equation

A circle is a unique geometric shape defined in a coordinate system by an equation. The general equation form for a circle is \[ (x - h)^2 + (y - k)^2 = r^2 \] Here, \( (h, k) \) are the coordinates of the circle's center, and \( r \) is the radius. This formula is derived from the Pythagorean theorem and essentially represents all the points that maintain a constant distance (the radius) from a central point (the center).
When plotting a circle on a coordinate plane, this equation helps establish its exact position and size in the 2D space. The difference \( x - h \) indicates the shift along the horizontal axis, and \( y - k \) reflects the shift vertically. By substituting the known values, like in the case of the Cosmo Clock 21 Ferris wheel, you determine the specific and precise location of any circle given its key features.

Radius Calculation

To find the radius of a circle, you divide its diameter by 2. This is because the diameter is simply twice the length of the radius since it stretches across the entire circle through the center. Using this simple formula:\[ r = \frac{d}{2} \]You can easily calculate the radius if you know the diameter. For the Cosmo Clock 21, the diameter is 100 meters, so its radius is:

• The radius is \( 50 \) meters.

Knowing the radius is crucial not only for drawing the circle but also for determining various attributes, such as area or circumference. Whenever you need to understand circular distances and measurements, always consider the radius your most vital starting point in calculations.

Coordinate System

Coordinate systems play a crucial role in geometry, especially when dealing with the placement and manipulation of shapes like circles. A common coordinate system is the Cartesian plane, where each point is identified by a pair of numbers, usually \(x\) and \(y\). These coordinates allow for precise plotting of points and shapes.For the Ferris wheel, determining the coordinates of its center involved recognizing the vertical and horizontal positions.

• The vertical component, namely the height from the ground to its center, is crucial.
• The horizontal component reflects its position from a center point, often taken as 0 unless otherwise defined.

For the Cosmo Clock 21, the center coordinates were determined as \((0, 55)\), with 55 meters being its vertical height above the base level. Understanding coordinates helps in accurately applying the circle equation and is fundamental in geometry for mapping out any object or shape in a 2D space.