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 use the information given about the direct variation of \( x \) with \( (y^2 + z^2) \).

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

### Step 2: Find the value of \( k \)
We know that when \( y = 1 \) and \( z = 2 \), \( x = 15 \). 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 substitute back into the equation:
\[ 15 = k(5) \]
To find \( k \), 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 \( y = 2 \). Substitute these values into the equation:
\[ 39 = 3(2^2 + z^2) \]
Calculating \( 2^2 \):
\[ 2^2 = 4 \]
Now substitute this back into the equation:
\[ 39 = 3(4 + z^2) \]
### Step 4: Simplify the equation
Divide both sides by 3:
\[ 13 = 4 + z^2 \]
Subtract 4 from both sides:
\[ 13 - 4 = z^2 \]
\[ 9 = z^2 \]

### Step 5: Solve for \( z \)
To find \( z \), take the square root of both sides:
\[ z = \sqrt{9} \]
Thus, we have:
\[ z = 3 \]

### Final Answer
The value of \( z \) when \( x = 39 \) and \( y = 2 \) is \( z = 3 \).

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