Problem 48 - Thomas Calculus

Question

Find the centers and radii of the spheres in Exercises \(47-50\) $$(x-1)^{2}+\left(y+\frac{1}{2}\right)^{2}+(z+3)^{2}=25$$

Step-by-Step Solution

Answer

The center is \\( (1, -\frac{1}{2}, -3) \\) and the radius is \\( 5 \\).

Step 1: Understand the Sphere Equation

The general equation of a sphere in three dimensions is given by \[(x-h)^2 + (y-k)^2 + (z-l)^2 = r^2\]where \(h, k, l\) represents the center of the sphere, and \(r\) is the radius.

Step 2: Identify the Center (h, k, l)

Comparing the given equation \[(x-1)^2 + \left(y + \frac{1}{2}\right)^2 + (z+3)^2 = 25\] with the general form, we can identify the center of the sphere as:- \(h = 1\) from \(x - 1\)- \(k = -\frac{1}{2}\) from \(y + \frac{1}{2}\)- \(l = -3\) from \(z + 3\).Thus, the center is \( (1, -\frac{1}{2}, -3) \).

Step 3: Determine the Radius r

In the given form, the right side of the equation equals the square of the radius:\[r^2 = 25\]By taking the square root of both sides:\[r = \sqrt{25} = 5\]Thus, the radius \(r\) is 5.

Key Concepts

Understanding the Center of a Sphere

The center of a sphere is a vital component in its mathematical equation. It helps describe the sphere's precise location in three-dimensional space.
The sphere's equation is typically expressed as \[(x-h)^2 + (y-k)^2 + (z-l)^2 = r^2.\] Here,

• \(h\) represents the x-coordinate of the center,
• \(k\) represents the y-coordinate,
• \(l\) represents the z-coordinate.

To determine the center from an equation, compare it to the general form.
For the equation provided:\[(x-1)^2 + \left(y + \frac{1}{2}\right)^2 + (z+3)^2 = 25.\]By matching terms, we identify that:

• \(h = 1\),
• \(k = -\frac{1}{2}\),
• \(l = -3\).

Therefore, the center of this sphere is located at \((1, -\frac{1}{2}, -3)\). This point signifies the sphere's middle point in 3D space, from which all surface points are equidistant.

Determining the Radius of a Sphere

The radius of a sphere is the distance from its center to any point on its surface. It plays a crucial role in defining the sphere's size.
In the equation \[(x-h)^2 + (y-k)^2 + (z-l)^2 = r^2,\]\(r^2\) represents the square of the radius.To calculate the radius, we need to solve for \(r\) by taking the square root of the right-hand side of the equation.
For our given equation: \[(x-1)^2 + \left(y + \frac{1}{2}\right)^2 + (z+3)^2 = 25,\]it's evident that:

• \(r^2 = 25\)

To find \(r\), we compute:\[r = \sqrt{25} = 5.\]Hence, the sphere has a radius of 5 units, meaning all points on the sphere's surface are 5 units away from the center.

Exploring Three-Dimensional Geometry

Three-dimensional geometry allows us to model and understand the universe with depth, width, and height. One of its essential objects is the sphere, a perfect 3D shape with symmetry from all points of its center.
In 3D space, every object or shape is defined by its position and size. For spheres, these aspects are determined by:

• The center: which gives its exact location in space,
• The radius: which defines how large or small it is.

Three-dimensional equations enable accurate modeling of these shapes, helping us visualize spheres mathematically. Each equation provides all the necessary information to recreate the shape and understand its interactions with other objects in space. This understanding is pivotal in fields like physics, engineering, and computer graphics, where spatial visualization is key.