Question

The surface area of the sphere \(x^{2}+y^{2}+z^{2}+2 x-4 y+8 z-2=0\) is , .....

Short answer

The surface area of the sphere is \(92\pi\).

Step-by-step solution

Identify the Equation of a Sphere

The general form of a sphere's equation is \( (x-h)^2 + (y-k)^2 + (z-l)^2 = r^2 \) where \((h, k, l)\) is the center and \(r\) is the radius. The given equation is \( x^2 + y^2 + z^2 + 2x - 4y + 8z - 2 = 0 \).

Complete the Square for each Term

Complete the square for each of the variables:- For \(x\), add and subtract \((\frac{2}{2})^2 = 1\): \((x+1)^2 - 1\)- For \(y\), add and subtract \((-\frac{4}{2})^2 = 4\): \((y-2)^2 - 4\)- For \(z\), add and subtract \((\frac{8}{2})^2 = 16\): \((z+4)^2 - 16\).

Formulate the Completed Equation

Substitute these into the original equation:\( (x+1)^2 - 1 + (y-2)^2 - 4 + (z+4)^2 - 16 = 2 \)Simplifying yields:\( (x+1)^2 + (y-2)^2 + (z+4)^2 = 23 \).

Identify Center and Radius

The equation \( (x+1)^2 + (y-2)^2 + (z+4)^2 = 23 \) shows the sphere with center \((-1, 2, -4)\) and radius \( r = \sqrt{23} \).

Calculate Surface Area of the Sphere

The formula for the surface area of a sphere is \(4\pi r^2\). Substitute \( r = \sqrt{23} \):\[ 4\pi (\sqrt{23})^2 = 4\pi \times 23 = 92\pi. \]

Conclude the Total Surface Area

The calculated surface area of the sphere is \(92\pi\).

Key concepts

Completing the Square

Completing the square is a mathematical technique used to make quadratic expressions easier to work with. This process rewrites a quadratic equation in a way that involves perfect squares, which becomes super handy when deriving the equation of a sphere from its expanded form. Here's a simple way to visualize this:

• For a term like \(x^2 + 2x\), you aim to express it as \((x+a)^2 - a^2\). By calculating \(a\) as half of the linear term's coefficient, you get the perfect square trinomial.
• For \(x^2 + 2x\), half of 2 is 1. So, thus it becomes \((x+1)^2 - 1\).
• This technique is repeated similarly for \(y^2\) and \(z^2\) terms. It helps you restructure the equation by moving constants on one side and grouping variables into perfect squares.

After completing the squares for each variable, the messy set of terms transforms into a much more manageable sphere equation, setting the foundation for identifying the sphere's center and radius.

Equation of a Sphere

The equation of a sphere is essentially a 3D extension of a circle's equation. In its canonical form, it is expressed as \[(x-h)^2 + (y-k)^2 + (z-l)^2 = r^2\]where

• \((h, k, l)\) represents the center of the sphere in a 3-dimensional space,
• \(r\) is the radius.

In our exercise, after we complete the squares, the originally complicated equation \[ (x^2 + y^2 + z^2 + 2x - 4y + 8z - 2 = 0) \]transforms into \[ (x+1)^2 + (y-2)^2 + (z+4)^2 = 23 \].This tells us:

• The sphere is centered at \((-1, 2, -4)\).
• The radius squared \(r^2\) is equal to 23, helping us know about its size.

Understanding this equation form is crucial for visualizing spheres in space and performing further calculations like determining the surface area.

Sphere Radius Calculation

Calculating the radius of a sphere from its equation involves determining the distance from its center to any point on its surface. Once you have the equation in the form \[(x-h)^2 + (y-k)^2 + (z-l)^2 = r^2\],the radius \(r\) can be found by taking the square root of the number on the right side of the equation.For the provided problem,

• The equation we derived is \((x+1)^2 + (y-2)^2 + (z+4)^2 = 23\)
• Here, \(r^2 = 23\).
• So, the radius \(r\) would be \(\sqrt{23}\).

This \(\sqrt{23}\) forms the basis for calculating the sphere's surface area. Using the formula \(4\pi r^2\), the surface area is easily determined when you know the radius, providing insight into the sphere's geometry.