Question

A spherical building has a diameter of 165 feet. The center of the building is placed at the origin of a three-dimensional coordinate system. What is the equation of the sphere?

Short answer

The equation of the sphere with a diameter of 165 feet, centered at the origin of a three-dimensional coordinate system is \(x^2 + y^2 + z^2 = 6806.25\).

Step-by-step solution

Identifying the radius

Given the diameter is 165 feet, the radius is half of this value. Hence, the radius = 165 / 2 = 82.5 feet.

Plug In the Radius

Substitute this radius into the equation of a sphere centered at the origin, which is \(x^2 + y^2 + z^2 = r^2\). This results in \(x^2 + y^2 + z^2 = 82.5^2\).

Simplify the Equation

Squaring 82.5, we get 6806.25. Therefore, the equation of the sphere turns into \(x^2 + y^2 + z^2 = 6806.25\).

Key concepts

Three-dimensional coordinate system

When we talk about spheres and their positioning in space, we use a three-dimensional coordinate system. This system is a method to determine the location of points. Think of it as a more advanced version of the regular graph you use in mathematics. However, instead of just having an x-axis and a y-axis, we add a z-axis. This third axis allows us to locate points in three dimensions: up-and-down, left-and-right, and forward-and-backward.

For a sphere, its position in this three-dimensional space is defined by its center point. A sphere can be moved anywhere in space by changing the coordinates of this center. In our exercise, the center is placed at the origin. The origin is the point \(0, 0, 0\), where all three axes intersect. Placing the sphere's center at the origin makes calculations simpler. Everything is then symmetrical around this point.

This system is essential for understanding many practical applications like architecture, engineering, and computer graphics. By mastering this, students can begin to visualize and analyze complex spatial structures.

Spherical geometry

Spherical geometry deals with the properties and relationships of points, lines, and surfaces on a sphere. Unlike the flat surfaces in regular geometry, spherical surfaces curve. This curvature introduces interesting properties, such as the fact that the shortest path between two points is an arc, not a straight line.

In our specific exercise, you are learning about spheres through the equation of a sphere. The basic equation to represent a sphere centered at the origin is \(x^2 + y^2 + z^2 = r^2\). This equation is derived from the essential formula of a radius in three-dimensional space, where \(x, y,\) and \(z\) are the distances from the center of the sphere in each respective direction. This formula allows us to model spheres precisely in the three-dimensional coordinate system.

Applications of spherical geometry extend beyond mathematics. It applies to astronomy, where planets and stars are considered spheres, and even to computer graphics, where curved surfaces are essential.

Diameter and radius calculations

Understanding the relationship between diameter and radius is crucial for working with spheres. The diameter of a sphere is the longest distance across it, passing through the center. In simple terms, it is the 'width' of the sphere at its widest point.

The radius is half of the diameter. It stretches from the center of the sphere to any point on its surface. In our exercise example, the given diameter is 165 feet. Therefore, the calculation to find the radius is straightforward: divide the diameter by two.

• Given diameter: 165 feet
• Radius calculation: \(\frac{165}{2} = 82.5 \text{ feet}\)

These calculations are vital for designing and building spherical structures. They translate straightforwardly into the sphere's mathematical model and the spherical equation. Knowing these measurements ensures your calculations are accurate and can be further used in engineering and construction plans.