Problem 11 - Essential Calculus Early Transcendentals

Question

Find an equation of the sphere that passes through the point \((4,3,-1)\) and has center \((3,8,1) .\)

Step-by-Step Solution

Answer

The equation of the sphere is \\( (x - 3)^2 + (y - 8)^2 + (z - 1)^2 = 30. \\)

Step 1: Understand the Equation of a Sphere

The equation of a sphere with center \(h, k, l\) and radius \r\ is: \[ (x - h)^2 + (y - k)^2 + (z - l)^2 = r^2. \] In this exercise, the center is given as \(3, 8, 1\). So, the equation will be: \[ (x - 3)^2 + (y - 8)^2 + (z - 1)^2 = r^2. \] We need to find the radius \r, which is the distance from the center to the given point.

Step 2: Calculate the Radius of the Sphere

To determine \r, we use the distance formula between the center of the sphere \(3, 8, 1\) and the point \(4, 3, -1\). The distance, or radius, \r\ is calculated as: \[ r = \sqrt{(4 - 3)^2 + (3 - 8)^2 + (-1 - 1)^2}. \] Simplifying: \[ r = \sqrt{1^2 + (-5)^2 + (-2)^2} = \sqrt{1 + 25 + 4} = \sqrt{30}. \] Thus, \r^2 = 30.\

Step 3: Write the Equation of the Sphere

Substitute the center \(h = 3, k = 8, l = 1\) and \r^2 = 30\ into the sphere's equation form. Therefore, the equation of the sphere is: \[ (x - 3)^2 + (y - 8)^2 + (z - 1)^2 = 30. \]

Key Concepts

Understanding the Center of a Sphere

The center of a sphere is a point in three-dimensional space that is equidistant from every point on the surface of the sphere. In other words, if you were to trace from the center to any point on the sphere, the length of that line would remain constant. This center is represented by the coordinates \(h, k, l\).

• The center is crucial because it helps define the position of the sphere in space.
• The equation of a sphere uses this point to express distances from the center to the edge of the sphere.

In our given exercise, the center of the sphere is \(3, 8, 1\). This means that the sphere is positioned such that \(x=3, y=8,\) and \(z=1\) are its central coordinates. Understanding the center helps to effectively apply the formula for the equation of a sphere.

Defining the Radius of a Sphere

The radius of a sphere is the distance from its center to any point on its surface. It is a key element in determining the size of the sphere. In the equation of a sphere, the radius is denoted by \r\ and is squared.

• The larger the radius, the bigger the sphere.
• Once the radius is known, you can easily write the complete equation of the sphere.

For the given exercise, the radius is calculated using a specific point on the surface, \(4, 3, -1\). By finding the distance from this point to the center \(3, 8, 1\), we determine the radius. In our exercise, this distance is found to be \sqrt{30}\. Consequently, the value of \r^2\ becomes 30, which is then used to complete the sphere's equation.

Using the Distance Formula

The distance formula is an essential mathematical tool used to find the distance between two points in space. For three-dimensional space, the distance between two points \( (x_1, y_1, z_1) \) and \( (x_2, y_2, z_2) \) is given by: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \] Using the distance formula allows us to determine how far apart two points are in a 3D space.

• This is important for calculating the radius of a sphere when given a center point and another point on its surface.
• Once the radius is calculated, it can be squared and applied to the equation of the sphere.

In our example, using the distance formula between the point \(4, 3, -1\) and the center \(3, 8, 1\), we arrive at a radius of \sqrt{30}\, perfectly illustrating how the distance formula aids in solving the problem.