Problem 46 - College Algebra and Calculus: An Applied Approach

Question

Find the sphere's center and radius. $$ 4 x^{2}+4 y^{2}+4 z^{2}-8 x+16 y+11=0 $$

Step-by-Step Solution

Answer

The sphere's center is at (1, -2, 0) and the radius is \(\sqrt{6}\).

Step 1: Rewrite the equation

First, rewrite the equation, grouping the terms that contain the same variables: \(4(x^{2}-2x)+4(y^{2}+4y)+4(z^{2})= \(-11\)\)

Step 2: Complete the squares

Complete the squares for the expression grouped by variables. This is done by adding and subtracting the square of half the coefficients of x and y inside the brackets: \(4[(x-1)^{2}-1]+4[(y+2)^{2}-4]+4(z^{2})= -11\)

Step 3: Simplify the expression

Simplify the expression by expanding the brackets and moving the constant term to the right side of the equation: \((x-1)^{2}+(y+2)^{2}+z^{2}= \(-1 - 4 + 11 \) on calculating the right side we get \((x-1)^{2}+(y+2)^{2}+z^{2} = 6\)

Step 4: Identify the center and the radius

Now the equation is in its standard form. Using the general formula for a sphere's equation \((x-a)^{2}+(y-b)^{2}+(z-c)^{2}=r^{2}\), where the center is \((a, b, c)\) and \(r\) is the radius. we can equate these to find that the center of the sphere is \((1, -2, 0)\) and the radius is \(\sqrt{6}\)