Problem 37 - Essential Calculus Early Transcendentals

Question

Find the distance between the spheres \(x^{2}+y^{2}+z^{2}=4\) and \(x^{2}+y^{2}+z^{2}=4 x+4 y+4 z-11.\)

Step-by-Step Solution

Answer

The spheres overlap, so there is no distance between them in the separation sense.

Step 1: Identify the centers of the spheres

The equation of the first sphere is \(x^2 + y^2 + z^2 = 4\). This can be rewritten as \((x-0)^2 + (y-0)^2 + (z-0)^2 = 2^2\), indicating that the center is at \((0, 0, 0)\) and the radius is 2.The equation of the second sphere is \(x^2 + y^2 + z^2 = 4x + 4y + 4z - 11\), which can be rewritten as \((x-2)^2 + (y-2)^2 + (z-2)^2 = 3^2\), indicating that the center is at \((2, 2, 2)\) and the radius is 3.

Step 2: Calculate the distance between the centers of the spheres

Find the distance between the centers \((0, 0, 0)\) and \((2, 2, 2)\). Use the distance formula: \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}\).Substituting the coordinates, we have: \[\sqrt{(2 - 0)^2 + (2 - 0)^2 + (2 - 0)^2} = \sqrt{4 + 4 + 4} = \sqrt{12} = 2\sqrt{3}.\]

Step 3: Determine the separation between the spheres

The spheres will be separate if the distance between their centers is greater than the sum of their radii. The sum of the radii is \(2 + 3 = 5\).Since \(2\sqrt{3} \approx 3.464\), the distance between the centers is less than 5, which means the spheres are overlapping and thus we can't find a separation distance in the mathematical sense.

Key Concepts

Equation of a Sphere

The equation of a sphere in 3-dimensional space is a simple yet powerful concept. It is expressed in the form:
\((x - h)^2 + (y - k)^2 + (z - l)^2 = r^2\).
This equation defines all the points \(x, y, z\) that lie on the surface of a sphere with:

• Center at \(h, k, l\)
• Radius equal to \(r\)

Each term \((x - h)^2, (y - k)^2, (z - l)^2\) represents the squares of the differences between the coordinates of any point on the sphere and the center of the sphere.
For instance, in the problem at hand, the first sphere's equation is \(x^2 + y^2 + z^2 = 4\). This is equivalent to \((x - 0)^2 + (y - 0)^2 + (z - 0)^2 = 2^2\).
This reveals that the center of this sphere is at \(0, 0, 0\) and the radius is 2.
A sphere's equation is fundamental because it not only gives us geometric insight but serves as the foundation for more complex geometrical computations, such as finding distances and relative positions.

Distance Formula

The distance formula is a crucial tool in coordinate geometry, especially useful when dealing with spatial figures like spheres. It helps calculate the distance between two points in space:
\[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}\]
This takes into consideration the squared differences of corresponding coordinates, which are then summed and square-rooted to provide the Euclidean distance.

• \((x_1, y_1, z_1)\) are the coordinates of the first point.
• \((x_2, y_2, z_2)\) are the coordinates of the second point.

In our exercise, this formula is employed to find the distance between the centers of two spheres with centers \(0, 0, 0\) and \(2, 2, 2\).
Substituting into the distance formula gives \[\sqrt{(2 - 0)^2 + (2 - 0)^2 + (2 - 0)^2} = \sqrt{12} = 2\sqrt{3}\].
Understanding this formula is essential for comparing spatial relationships in geometry.

Center of a Sphere

The center of a sphere is key to understanding its spatial properties. It effectively describes the point in 3-dimensional space that equidistantly aligns with all surface points of the sphere.
Finding the center is often crucial in problems involving spherical objects, such as determining distances or interactions between spheres.
To identify the center of a sphere from its equation, inspect the terms in the expression \(x - h)^2 + (y - k)^2 + (z - l)^2\). The coordinates \(h, k, l\) define the center. For example, in the given problem:

• For the first sphere with equation \((x-0)^2 + (y-0)^2 + (z-0)^2 = 2^2\), the center is \(0, 0, 0\).
• For the second sphere rewritten as \((x-2)^2 + (y-2)^2 + (z-2)^2 = 3^2\), the center is \(2, 2, 2\).

This information is pivotal, as it allows us to apply formulas like the distance formula to explore the spatial relationships further. Recognizing and interpreting the center of a sphere from its equation simplifies a multitude of geometric problems.