Problem 11 - Calculus Early Transcendentals: Pearson New International Edition

Question

Write the equation of the sphere with the given center and radius. (a) \((1,2,3) ; 5\) (b) \((-2,-3,-6) ; \sqrt{5}\) (c) \((\pi, e, \sqrt{2}) ; \sqrt{\pi}\)

Step-by-Step Solution

Answer

(a) \((x-1)^2 + (y-2)^2 + (z-3)^2 = 25\); (b) \((x+2)^2 + (y+3)^2 + (z+6)^2 = 5\); (c) \((x-\pi)^2 + (y-e)^2 + (z-\sqrt{2})^2 = \pi\)."

Step 1: Understand the Equation of a Sphere

The general equation of a sphere with center \((h, k, l)\) and radius \(r\) is given by \((x-h)^2 + (y-k)^2 + (z-l)^2 = r^2\). This defines all points \((x, y, z)\) that are at distance \(r\) from the center \((h, k, l)\).

Step 2: Write the Equation for Part (a)

For the center \((1, 2, 3)\) and radius \(5\), plug \(h = 1\), \(k = 2\), \(l = 3\), and \(r = 5\) into the equation. The equation is:\[(x - 1)^2 + (y - 2)^2 + (z - 3)^2 = 5^2\] which simplifies to:\[(x - 1)^2 + (y - 2)^2 + (z - 3)^2 = 25\]

Step 3: Write the Equation for Part (b)

For the center \((-2, -3, -6)\) and radius \(\sqrt{5}\), use \(h = -2\), \(k = -3\), \(l = -6\), and \(r = \sqrt{5}\). The equation becomes:\[(x + 2)^2 + (y + 3)^2 + (z + 6)^2 = (\sqrt{5})^2\] simplifying to:\[(x + 2)^2 + (y + 3)^2 + (z + 6)^2 = 5\]

Step 4: Write the Equation for Part (c)

For the center \((\pi, e, \sqrt{2})\) and radius \(\sqrt{\pi}\), substitute \(h = \pi\), \(k = e\), \(l = \sqrt{2}\), and \(r = \sqrt{\pi}\). The equation is:\[(x - \pi)^2 + (y - e)^2 + (z - \sqrt{2})^2 = (\sqrt{\pi})^2\] which simplifies to:\[(x - \pi)^2 + (y - e)^2 + (z - \sqrt{2})^2 = \pi\]

Key Concepts

Center of a sphere

The center of a sphere is a crucial component for defining the equation of the sphere. Imagine a perfect three-dimensional circle, or a ball. The center is the exact middle point inside that ball where all points on the surface are equidistant. It's like the bullseye in a dartboard, except in three dimensions. In mathematical terms, specifying the center of a sphere involves identifying a point with coordinates

• Coordinates:

  This is normally expressed as

  • (h, k, l), where
  • h is the x-coordinate,
  • k is the y-coordinate,
  • l is the z-coordinate.

  It's essential to correctly identify this point because it's the reference for measuring the radius and determining other points on the sphere.

  • Practical Importance:

    The center of the sphere allows us to clearly communicate the sphere's position in space.

    In our exercise examples, we were given centers like (1, 2, 3),(-2, -3, -6), and (π, e, √2).

    It’s essential to find this point accurately to write the equation of the sphere correctly.

  This understanding helps you effectively use formulas or equations that involve spheres, such as in geometry or physics problems.

Radius of a sphere

The radius of a sphere is another fundamental aspect when discussing its equation. The radius is simply the distance from the center of the sphere to any point on its surface. Think of this as the length from the microphone within a karaoke machine to the speaker's edge on the outside.

• Measurement

  This measurement is always a positive value and is constant across the surface.

  It defines the size of the sphere.

Using the radius in mathematical problems helps identify how large or small the sphere is relative to the center.

• Using the Radius in Equations

  In our sphere equation

  • (x-h)^2 + (y-k)^2 + (z-l)^2 = r^2,
  • r represents the radius squared.

  Thus, we need to square the radius when substituting into the formula as such

  If the radius given is √5, as in our part (b), we use (√5)^2 which equals 5.

  Each equation must exactly reflect the given radius to accurately depict the sphere's characteristics.

Sphere equation derivation

Deriving the equation of a sphere seamlessly combines the ideas of the center and radius. This operation helps in forming a mathematical representation of a sphere using its center and radius.

• General Sphere Equation

  Understand that the general equation is a powerful tool:

  • (x-h)^2 + (y-k)^2 + (z-l)^2 = r^2.
  • Each component of the formula is vital:
    • h, k, and l, from the center, shift the origin point of the sphere’s location
    • r dictates the size of the sphere
    • By setting an equation, it explicitly defines every point on the surface of the sphere by using the distance from the center.

    Practical Application in the Exercise

    Let's examine the step-by-step construction using examples from the exercise:

    • When we have the center at (1, 2, 3) with a radius of 5, the equation (x-1)^2 + (y-2)^2 + (z-3)^2 = 25 offers a complete depiction.
    • Similarly, utilizing different centers and radii must strictly include these into the established equation format.

    This consistency ensures any sphere's appropriate graphical formulation in space, making analysis or projection feasible.