Problem 15 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 13-16, complete the squares to find the center and \(\mathrm{ra}\) dius of the sphere whose equation is given (see Example 2). $$ 4 x^{2}+4 y^{2}+4 z^{2}-4 x+8 y+16 z-13=0 $$

Step-by-Step Solution

Answer

Center: \((\frac{1}{2}, -1, -2)\), Radius: \(\frac{\sqrt{37}}{2}\)."

Step 1: Simplify the Equation

First, we divide every term in the equation by 4 to simplify it. This gives us: \(x^2 + y^2 + z^2 - x + 2y + 4z - rac{13}{4} = 0\).

Step 2: Rearrange and Group Terms

Rearrange the terms to separate the constant from the variable terms, grouping the \(x\), \(y\), and \(z\) terms: \(x^2 - x + y^2 + 2y + z^2 + 4z = \frac{13}{4}\).

Step 3: Completing the Square for x

For the \(x\) terms: \(x^2 - x\), we complete the square by taking half of the coefficient of \(x\), \(-\frac{1}{2}\), squaring it, and adding & subtracting it inside the equation: \((x - \frac{1}{2})^2 - \frac{1}{4}\).

Step 4: Completing the Square for y

For the \(y\) terms: \(y^2 + 2y\), complete the square by taking half of the coefficient of \(y\), adding & subtracting it: \((y + 1)^2 - 1\).

Step 5: Completing the Square for z

For the \(z\) terms: \(z^2 + 4z\), complete the square by adding & subtracting: \((z + 2)^2 - 4\).

Step 6: Rewrite the Equation

Substitute the completed square forms back into the equation: \((x - \frac{1}{2})^2 - \frac{1}{4} + (y + 1)^2 - 1 + (z + 2)^2 - 4 = \frac{13}{4}\).

Step 7: Combine Constants and Format

Combine all constants on the left to find the correct format for a sphere equation: \((x - \frac{1}{2})^2 + (y + 1)^2 + (z + 2)^2 = \frac{13}{4} + \frac{1}{4} + 1 + 4\).

Step 8: Solve for Center and Radius

Calculate the right side to get: \(\frac{37}{4}\). Thus, the sphere is centered at \((\frac{1}{2}, -1, -2)\) with radius \(\sqrt{\frac{37}{4}} = \frac{\sqrt{37}}{2}\).

Key Concepts

Equation of a Sphere

The equation of a sphere in the simplest form is \((x - h)^2 + (y - k)^2 + (z - l)^2 = r^2\), where:

• \( (h, k, l) \) is the center of the sphere.
• \( r \) is the radius of the sphere.

This equation emerges when identifying all points that are a fixed distance, \( r \),from a specific point, the center \((h, k, l)\).
Completing the square is a key technique in bringing a given equation into this standard sphere form. Without doing this,recognizing the center and radius is not straightforward, especially if the equation has mixed or linear terms.
With practice, you'll be able to spot opportunities to apply this method and transform complex equations with ease.

Mathematical Problem Solving

Mathematical problem solving often involves transforming or manipulating the equations we are given into a more recognizable form.
The process often requires us to break down the problem into simpler parts or steps that are easier to handle.In the context of this sphere problem, the goal is to identify the center and radius of the sphere from a given quadratic equation.

• The first step in this type of problem is to simplify the equation, making it easier to work with. Often, you divide all terms to clear coefficients.
• Next, rearrange terms to prepare for completing the square.
• Each variable, \(x\), \(y\), and \(z\), is treated separately, transforming each quadratic term into a reducible square form.

Mastering this problem-solving technique not only aids in comprehending geometric figures like spheres but also sharpens an overall mathematical skillset.

Algebraic Manipulation

Algebraic manipulation is a powerful tool required for transforming expressions and solving equations. It involves rearranging, simplifying, or factoring terms to achieve a desired outcome.
In this exercise, we focus on completing the square, a specific type of manipulation used to restructure quadratic terms
Here's a quick guide on how to complete the square:

• Identify the quadratic and linear terms of a single variable, say \(x^2 - ax\).
• Find half of the linear coefficient: here, it's \(-\frac{a}{2}\).
• Square this value: \((\frac{a}{2})^2\) and add & subtract it within the equation to maintain equivalence.
• Rewrite the expression as a perfect square, that reveals the inherent structure: \((x - \frac{a}{2})^2\) minus the squared term.

Using these steps, solve each variable term's quadratics to arrive at a neat equation.Algebraic manipulation helps in converting the challenge of a complex-looking equation into a straightforward, understandable form.