Problem 38 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the equations of the tangent spheres of equal radii whose centers are \((-3,1,2)\) and \((5,-3,6)\).

Step-by-Step Solution

Answer

The equations are \((x + 3)^2 + (y - 1)^2 + (z - 2)^2 = 24\) and \((x - 5)^2 + (y + 3)^2 + (z - 6)^2 = 24\).

Step 1: Calculate the Distance Between Centers

To find the distance between the centers of the spheres, use the distance formula for points in 3D space: \[D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}\]Substitute \((x_1, y_1, z_1) = (-3, 1, 2)\) and \((x_2, y_2, z_2) = (5, -3, 6)\):\[D = \sqrt{(5 + 3)^2 + (-3 - 1)^2 + (6 - 2)^2} = \sqrt{8^2 + (-4)^2 + 4^2}\]Calculate:\[D = \sqrt{64 + 16 + 16} = \sqrt{96} = 4\sqrt{6}\]

Step 2: Determine the Radius of the Spheres

Since the spheres are tangent and have equal radii, each sphere's radius is half the distance between the centers:\[R = \frac{4\sqrt{6}}{2} = 2\sqrt{6}\]

Step 3: Write the Equations of the Spheres

To write the equations of the spheres, use the standard sphere equation centered at \((a, b, c)\) with radius \(R\):\[(x - a)^2 + (y - b)^2 + (z - c)^2 = R^2\]For the first sphere centered at \((-3, 1, 2)\):\[(x + 3)^2 + (y - 1)^2 + (z - 2)^2 = (2\sqrt{6})^2 = 24\]For the second sphere centered at \((5, -3, 6)\):\[(x - 5)^2 + (y + 3)^2 + (z - 6)^2 = (2\sqrt{6})^2 = 24\]

Key Concepts

Distance Formula in 3D Space

The distance formula is a crucial tool in 3D geometry for determining the straight-line distance between two points in space. It's an extension of the 2D distance formula and accounts for the three dimensions by incorporating the coordinates in the x, y, and z axes. When finding the distance between two points, such as \((-3, 1, 2)\) and \(5, -3, 6\), we use the formula:

• \[ D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \]

Substituting the given values, the calculation becomes:

• \[ D = \sqrt{(5 + 3)^2 + (-3 - 1)^2 + (6 - 2)^2} \]
• \[ D = \sqrt{8^2 + (-4)^2 + 4^2} \]

The result of these calculations gives us the distance between the centers as \( 4\sqrt{6} \). Understanding how to use and apply this formula in 3D is foundational for solving problems involving distances between point coordinates in space.

Understanding 3D Geometry

3D geometry deals with spatial figures in a three-dimensional space, defined by three axes: x, y, and z. This realm of geometry is fundamental when dealing with objects like spheres, as it allows consideration of depth in addition to length and width. In 3D geometry, visualization becomes key. When working with spheres, you’re often dealing with objects that exist in all three planes described by their centers and radii. For instance, spheres have volume which is determined not just by their radii but also by their positioning in space. In exercises like the one addressed, recognizing how spheres relate in a shared space, whether tangentially or intersecting, is crucial. Moreover, understanding the plotting of points and their relation is vital when working with scenarios involving more than one sphere. Thus, mastering 3D geometry concepts opens avenues to solving complex spatial problems efficiently.

Sphere Equations Explained

The equation of a sphere in 3D space is derived from the distance formula and expresses the set of all points equidistant from a center point. This set of points is the surface of the sphere. The general equation of a sphere with center \( (a, b, c) \) and radius \( R \) is:

• \[ (x - a)^2 + (y - b)^2 + (z - c)^2 = R^2 \]

This formulation ensures that any coordinates \( (x, y, z) \) lying on the sphere's surface maintain a constant distance equal to \( R \) from the center \( (a, b, c) \).For example, given centers \((-3, 1, 2)\) and \(5, -3, 6\) and radii \(2\sqrt{6}\), we apply these into the sphere equation for each center to obtain:

• First Sphere: \[(x + 3)^2 + (y - 1)^2 + (z - 2)^2 = 24\]
• Second Sphere: \[(x - 5)^2 + (y + 3)^2 + (z - 6)^2 = 24\]

These equations represent each sphere fully, showcasing their set positions in 3D space.

Radii of Spheres: Role and Calculation

In the realm of spheres, the radius is the distance from the center to any point on the surface of the sphere. It directly influences both the size of the sphere and its spatial properties. When two spheres are tangent to each other and have equal radii, it simplifies our calculations to recognize that their cumulative radius equals the distance between their centers divided by two.For our exercise, the calculated distance between the centers is \(4\sqrt{6}\), thus each sphere will have a radius of:

• \[ R = \frac{4\sqrt{6}}{2} = 2\sqrt{6} \]

This calculation is fundamental to formulating the spheres’ equations correctly. It ensures each is constructed with a precise and equal spread from their respective centers, creating physical spheres that meet exactly at one point—the definition of tangent spheres.Understanding radii not only aids in comprehending the sphere's dimensions but also enhances our ability to solve spatial problems that revolve around spheres and their interactions.