Problem 15 - Precalculus Mathematics for Calculus

Question

Show that the equation represents a sphere, and find its center and radius. $$x^{2}+y^{2}+z^{2}-10 x+2 y+8 z=9$$

Step-by-Step Solution

Answer

The sphere's center is at (5, -1, -4) and its radius is \(\sqrt{51}\).

Step 1: Rearrange Terms

First, we need to group like terms for each variable in the equation \(x^{2} + y^{2} + z^{2} - 10x + 2y + 8z = 9\). This will help us in completing the square for each term.

Step 2: Complete the Square for x

For the \(x\) terms: \(x^{2} - 10x\). The term needed to complete the square is \(\left(\frac{-10}{2}\right)^2 = 25\). Add and subtract 25 inside the equation so it becomes \((x^{2} - 10x + 25 - 25)\). This simplifies to \((x - 5)^2 - 25\).

Step 3: Complete the Square for y

For the \(y\) terms: \(y^{2} + 2y\). The term needed to complete the square is \(\left(\frac{2}{2}\right)^2 = 1\). Add and subtract 1 inside the equation so it becomes \((y^{2} + 2y + 1 - 1)\). This simplifies to \((y + 1)^2 - 1\).

Step 4: Complete the Square for z

For the \(z\) terms: \(z^{2} + 8z\). The term needed to complete the square is \(\left(\frac{8}{2}\right)^2 = 16\). Add and subtract 16 inside the equation so it becomes \((z^{2} + 8z + 16 - 16)\). This simplifies to \((z + 4)^2 - 16\).

Step 5: Form the Complete Equation

Substitute the completed squares back into the equation: \((x - 5)^2 - 25 + (y + 1)^2 - 1 + (z + 4)^2 - 16 = 9\).Simplify this to:\((x - 5)^2 + (y + 1)^2 + (z + 4)^2 = 51\).

Step 6: Identify Center and Radius

Recognize the equation \((x - 5)^2 + (y + 1)^2 + (z + 4)^2 = 51\) as the standard form of a sphere, \((x-a)^2 + (y-b)^2 + (z-c)^2 = r^2\). Therefore, the center of the sphere is \((5, -1, -4)\) and the radius is \(\sqrt{51}\).

Key Concepts

Completing the Square

When tackling the equation of a sphere, completing the square becomes an essential tool. It's a method used to rearrange quadratic expressions into a perfect square trinomial, which clarifies the structure of equations. Here's how it helps:

Suppose you have a quadratic expression like \(x^2 - 10x\). The goal is to convert this into the form \((x - a)^2\). To achieve this, you first find the term that makes it a perfect square. You take the coefficient of \(x\), which is \(-10\), divide it by 2 to get \(-5\), and then square it to get 25. Hence, \((x^2 - 10x + 25) = (x - 5)^2\).

In the same way, complete the square for \(y^2 + 2y\) to get \((y + 1)^2\) and for \(z^2 + 8z\) to get \((z + 4)^2\). Completing these squares makes it easier to see the overall shape and center of the sphere.

Center of a Sphere

The center of a sphere is a critical concept in understanding its geometry. In the transformed equation of a sphere, represented in terms like \((x - 5)^2 + (y + 1)^2 + (z + 4)^2 = r^2\), the sphere's center can be directly read from these expressions.

Note the pattern for each variable where the expression \((x - a)^2\), \((y - b)^2\), and \((z - c)^2\) tells us that \(a\), \(b\), and \(c\) are the coordinates of the center. In this equation, they are 5, -1, and -4 respectively. Therefore, the center of this sphere is clearly \((5, -1, -4)\).

This method provides a straightforward way to identify the center, allowing for an easy analysis of the sphere's position in space.

Radius of a Sphere

The radius of a sphere is another core feature of its geometry, signifying the distance from the center to any point on its surface. In the equation \((x - 5)^2 + (y + 1)^2 + (z + 4)^2 = r^2\), the term on the right side, 51 in this case, is crucial.

In this format, \(r^2\) provides the square of the radius. To find the actual radius \(r\), you merely take the square root of this term. Therefore, the radius of the sphere here is \(\sqrt{51}\).

This understanding is critical when working with spheres, as the radius helps in determining the size and scale of the sphere as it relates to other objects or spaces.