x varies directly as `(y^(2)+z^(2))` At y=1 and z=2, the value of x is 15. Find the value of z when x=39 and y=2

Options

A. 2
B. 3
C. 4
D. 6

Answer

To solve the problem step by step, we will follow the direct variation relationship given in the question.

### Step 1: Establish the relationship
Since \( x \) varies directly as \( (y^2 + z^2) \), we can express this relationship mathematically as:
\[ x = k(y^2 + z^2) \]
where \( k \) is the constant of proportionality.

### Step 2: Find the constant \( k \)
We know that when \( y = 1 \) and \( z = 2 \), the value of \( x \) is 15. We can substitute these values into the equation to find \( k \):
\[ 15 = k(1^2 + 2^2) \]
Calculating \( 1^2 + 2^2 \):
\[ 1^2 + 2^2 = 1 + 4 = 5 \]
Now substituting back into the equation:
\[ 15 = k(5) \]
To find \( k \), we divide both sides by 5:
\[ k = \frac{15}{5} = 3 \]

### Step 3: Use the value of \( k \) to find \( z \)
Now we need to find the value of \( z \) when \( x = 39 \) and \( y = 2 \). We substitute these values into the equation:
\[ 39 = 3(2^2 + z^2) \]
Calculating \( 2^2 \):
\[ 2^2 = 4 \]
Substituting this into the equation gives:
\[ 39 = 3(4 + z^2) \]
Now, divide both sides by 3:
\[ 13 = 4 + z^2 \]
Next, we isolate \( z^2 \) by subtracting 4 from both sides:
\[ z^2 = 13 - 4 = 9 \]
Taking the square root of both sides gives:
\[ z = \sqrt{9} = 3 \]

### Final Answer
The value of \( z \) when \( x = 39 \) and \( y = 2 \) is:
\[ \boxed{3} \]