The radius of the sphere
`3x^2+3y^2+3z^2-8x+4y+8z-15=0` is

Options

A. 2
B. 3
C. 4
D. 5

Answer

To find the radius of the sphere given by the equation \(3x^2 + 3y^2 + 3z^2 - 8x + 4y + 8z - 15 = 0\), we can follow these steps:

### Step 1: Simplify the Equation
First, we can simplify the equation by dividing all terms by 3:
\[ x^2 + y^2 + z^2 - \frac{8}{3}x + \frac{4}{3}y + \frac{8}{3}z - 5 = 0 \]

### Step 2: Rearrange the Equation
Rearranging the equation gives us:
\[ x^2 + y^2 + z^2 - \frac{8}{3}x + \frac{4}{3}y + \frac{8}{3}z = 5 \]

### Step 3: Identify Coefficients
We can compare this with the standard form of a sphere's equation:
\[ x^2 + y^2 + z^2 + 2ux + 2vy + 2wz + d = 0 \]
From our equation, we can identify:
- \(2u = -\frac{8}{3}\) → \(u = -\frac{4}{3}\)
- \(2v = \frac{4}{3}\) → \(v = \frac{2}{3}\)
- \(2w = \frac{8}{3}\) → \(w = \frac{4}{3}\)
- \(d = -5\)

### Step 4: Calculate the Radius
The radius \(r\) of the sphere is given by the formula:
\[ r = \sqrt{u^2 + v^2 + w^2 - d} \]
Substituting the values we found:
\[ r = \sqrt{\left(-\frac{4}{3}\right)^2 + \left(\frac{2}{3}\right)^2 + \left(\frac{4}{3}\right)^2 - (-5)} \]

### Step 5: Simplify the Expression
Calculating each term:
- \(u^2 = \left(-\frac{4}{3}\right)^2 = \frac{16}{9}\)
- \(v^2 = \left(\frac{2}{3}\right)^2 = \frac{4}{9}\)
- \(w^2 = \left(\frac{4}{3}\right)^2 = \frac{16}{9}\)

Now, add these values:
\[ u^2 + v^2 + w^2 = \frac{16}{9} + \frac{4}{9} + \frac{16}{9} = \frac{36}{9} = 4 \]

Now, substituting back into the radius formula:
\[ r = \sqrt{4 + 5} = \sqrt{9} = 3 \]

### Final Answer
The radius of the sphere is \(3\).

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