Problem 14 - Essential Calculus Early Transcendentals

Question

Show that the equation represents a sphere, and find its center and radius. \(x^{2}+y^{2}+z^{2}+8 x-6 y+2 z+17=0\)

Step-by-Step Solution

Answer

The equation represents a sphere with center \((-4, 3, -1)\) and radius 3.

Step 1: Group the terms by variable

Start by grouping the terms that contain the same variable in the equation. This will prepare us to complete the square for each variable.The equation is \[ x^2 + 8x + y^2 - 6y + z^2 + 2z + 17 = 0 \]

Step 2: Complete the square for x

Take the terms involving \(x\), which are \(x^2 + 8x\). To complete the square, add and subtract \((\frac{8}{2})^2 = 16\).This gives us:\[ (x^2 + 8x + 16) - 16 \]Which becomes \[ (x + 4)^2 - 16 \]

Step 3: Complete the square for y

Now take the terms involving \(y\), which are \(y^2 - 6y\). To complete the square, add and subtract \((\frac{-6}{2})^2 = 9\).This gives us:\[ (y^2 - 6y + 9) - 9 \]Which becomes \[ (y - 3)^2 - 9 \]

Step 4: Complete the square for z

Take the terms involving \(z\), which are \(z^2 + 2z\). To complete the square, add and subtract \((\frac{2}{2})^2 = 1\).This gives us:\[ (z^2 + 2z + 1) - 1 \]Which becomes \[ (z + 1)^2 - 1 \]

Step 5: Substitute back into the original equation

Substitute the completed squares back into the original equation:\[(x + 4)^2 - 16 + (y - 3)^2 - 9 + (z + 1)^2 - 1 + 17 = 0\]Simplify the constant terms:\[ (x + 4)^2 + (y - 3)^2 + (z + 1)^2 - 9 = 0 \]Which can be written as:\[ (x + 4)^2 + (y - 3)^2 + (z + 1)^2 = 9 \]

Step 6: Identify the center and radius

The equation \((x + 4)^2 + (y - 3)^2 + (z + 1)^2 = 9\) is the standard form of a sphere's equation,\[ (x-a)^2 + (y-b)^2 + (z-c)^2 = r^2 \] where \((a, b, c)\) is the center and \(r\) is the radius. Here, the center is \((-4, 3, -1)\) and the radius is \(\sqrt{9} = 3\).

Key Concepts

Completing the Square

Completing the square is an algebraic technique used to transform a quadratic expression into a perfect square trinomial. This is useful for rewriting equations of spheres into a standard form, making it easier to identify centers and radii. Let's break it down with examples from our exercise:

• For the variable \(x\), the terms are \(x^2 + 8x\). To complete the square, find the term that, when both added and subtracted, turns it into a perfect square. The term is \((\frac{8}{2})^2 = 16\). Thus, \(x^2 + 8x\) becomes \((x + 4)^2 - 16\).
• For the variable \(y\), we have \(y^2 - 6y\). The necessary term is \((\frac{-6}{2})^2 = 9\), so it becomes \((y - 3)^2 - 9\).
• Finally, for \(z\), the expression is \(z^2 + 2z\). Here, the term is \((\frac{2}{2})^2 = 1\), which modifies it to \((z + 1)^2 - 1\).

The goal is to make each part of the quadratic into a "perfect square," to simplify the equation into the form of the sphere's standard equation: \((x-a)^2 + (y-b)^2 + (z-c)^2 = r^2\). This helps us easily recognize the center and radius of the sphere.

Center of a Sphere

The center of a sphere in a three-dimensional space is determined from its equation in standard form: \[(x-a)^2 + (y-b)^2 + (z-c)^2 = r^2\].Here, \(a\), \(b\), and \(c\) are the coordinates of the sphere's center. The exercise reveals this center after completing the squares for each variable.

In our example, the processed equation is: \[(x+4)^2 + (y-3)^2 + (z+1)^2 = 9\].From there, identify the shifts:

• For \(x\), the transformation was \((x+4)^2\), meaning a horizontal shift left by 4 units, so \(a = -4\).
• For \(y\), changing to \((y-3)^2\) means a vertical shift upwards by 3 units, resulting in \(b = 3\).
• Finally, \((z+1)^2\) reveals a backward shift by 1 unit, giving \(c = -1\).

Thus, the sphere's center is at \((-4, 3, -1)\). Knowing the center, you can easily visualize the exact placement of the sphere within a coordinate system.

Radius of a Sphere

The radius of a sphere is straightforward to determine once its equation is in the standard form\[(x-a)^2 + (y-b)^2 + (z-c)^2 = r^2\],where \(r\) represents the radius. To find the radius, take the square root of the equation's right side.

In this exercise, after simplifying the equation, we have:\[(x+4)^2 + (y-3)^2 + (z+1)^2 = 9\].Here, the expression equals 9, so to find the radius, compute:

• \(r^2 = 9\) leads to \(r = \sqrt{9}\), or \(r = 3\).

The radius being 3 means every point on the sphere is 3 units away from the center. It defines the sphere's size, and knowing the radius is vital for calculating the sphere's volume and surface area. Understanding this allows neater grappling of geometrical concepts tied to spheres in different dimensions.