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Magazine Chapter 7: Problem 6
The surface area of the sphere \(x^{2}+y^{2}+z^{2}+2 x-4 y+8 z-2=0\) is , .....
Short Answer
Expert verified
The surface area of the sphere is \(92\pi\).
Step by step solution
01
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 \( x^2 + y^2 + z^2 + 2x - 4y + 8z - 2 = 0 \).
02
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 \((\frac{8}{2})^2 = 16\): \((z+4)^2 - 16\).
03
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:\( (x+1)^2 + (y-2)^2 + (z+4)^2 = 23 \).
04
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 \( r = \sqrt{23} \).
05
Calculate Surface Area of the Sphere
The formula for the surface area of a sphere is \(4\pi r^2\). Substitute \( r = \sqrt{23} \):\[ 4\pi (\sqrt{23})^2 = 4\pi \times 23 = 92\pi. \]
06
Conclude the Total Surface Area
The calculated surface area of the sphere is \(92\pi\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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.
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.
- The sphere is centered at \((-1, 2, -4)\).
- The radius squared \(r^2\) is equal to 23, helping us know about its size.
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 \(\sqrt{23}\).
