Problem 92 - Intermediate Algebra

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 and completes a full rotation every 32 minutes. Its diameter is 150 meters, and the height of this giant wheel is 165 meters. 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 (d) to write an equation of the Singapore Flyer. (IMAGE CANNOT COPY)

Step-by-Step Solution

Answer

a. 75 meters b. 90 meters c. 165 meters d. \((0, 165)\) e. \(x^2 + (y-165)^2 = 5625\)

Step 1: Calculate the Radius

The diameter of the Singapore Flyer is given as 150 meters. To find the radius, use the formula for the radius, which is half of the diameter. Thus, the radius \( r \) is \( r = \frac{150}{2} = 75 \) meters.

Step 2: Determine How Close the Wheel is to the Ground

The height of the entire wheel structure is given as 165 meters. Since the radius of the wheel is 75 meters, and the center of the wheel is at the midpoint, we subtract the radius from the total height to find how far the bottom is from the ground: \( 165 - 75 = 90 \) meters.

Step 3: Calculate the Height of the Center of the Wheel from the Ground

The center of the wheel is located at the height of the radius from the base of the wheel structure. Since the bottom of the wheel is 90 meters from the ground and the radius is 75 meters, the center is at \( 90 + 75 = 165 \) meters above the ground.

Step 4: Determine the Coordinates of the Center

Assuming the ground level is \( y = 0 \) and the wheel is placed symmetrically around the vertical axis, the horizontal midpoint is zero. Therefore, the coordinates of the center of the wheel are \((0, 165)\).

Step 5: Write the Equation of the Wheel

The equation of a circle with a center at \((h, k)\) and radius \(r\) is \((x-h)^2 + (y-k)^2 = r^2\). Substituting \((h, k) = (0, 165)\) and \(r = 75\), the equation becomes \(x^2 + (y-165)^2 = 75^2\). Simplifying gives \(x^2 + (y-165)^2 = 5625\).

Key Concepts

Radius Calculation

When dealing with circles, one of the first things we often need to calculate is the radius. The radius is the distance from the center of the circle to any point on its circumference. In our exercise, we are given the diameter of the Singapore Flyer, which is 150 meters. You can find the radius using this simple relationship:

• The radius is half the diameter.
• Therefore, we calculate the radius as: \( r = \frac{150}{2} = 75 \) meters.

It's always crucial to remember that the radius is half of the diameter. This is a fundamental principle in geometry that applies to all circles, regardless of size.

Coordinate Geometry

In coordinate geometry, we often describe shapes by their positions on a plane using axes. For the Singapore Flyer, we determine the coordinates of its center. Given that the wheel is set symmetrically and the center is a critical point, we assume:

• The horizontal position, or 'x-coordinate', is zero.
• The vertical position, or 'y-coordinate', is the total height of its center from the ground, which is 165 meters.

Thus, the center of the Singapore Flyer wheel is at the coordinates \((0, 165)\). With circles, identifying the center coordinates allows for easy derivation of the circle's equation and assists in various geometric analyses.

Height Calculations

Height calculations are essential in determining positions relative to the ground. In our problem, we utilize height calculations to find out how close the wheel is to the ground and the absolute height of its center.

• The total height of the structure is 165 meters.
• The radius being the distance from the center to the lowest point is 75 meters.

To find how close the bottom edge of the wheel is to the ground, subtract the radius from the total height:
\[ 165 - 75 = 90 \text{ meters} \]This calculation shows that the lowest point of the wheel is 90 meters above the ground. Understanding this concept helps in comprehending the spatial placement of the wheel effectively.

Equation of a Circle

Creating an equation for a circle involves knowing the center and radius. For any circle, the general equation is:
\((x-h)^2 + (y-k)^2 = r^2\)Here, \((h, k)\) represents the circle's center and \(r\) its radius:

• In our example, the center is at \((0, 165)\).
• The radius is 75 meters.

Substituting these values, the equation becomes:
\[x^2 + (y-165)^2 = 75^2\]Simplifying gives us:
\[x^2 + (y-165)^2 = 5625\]This equation succinctly describes the Singapore Flyer's wheel, allowing for further analysis and understanding of its geometric properties.