What is the largest value of `m` for which there exists a real value of n such that `m^(2) = 196 -n^2`?

Options

A. 14
B. 95
C. 182
D. 196

Answer

To solve the problem, we need to find the largest value of \( m \) for which there exists a real value of \( n \) such that the equation \( m^2 = 196 - n^2 \) holds true. We can follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ m^2 = 196 - n^2 \]
Rearranging this gives:
\[ m^2 + n^2 = 196 \]

### Step 2: Analyze the equation
This equation represents a circle with a radius of \( \sqrt{196} = 14 \) in the \( m \)-\( n \) coordinate system. For \( n \) to be a real number, the expression \( 196 - m^2 \) must be non-negative.

### Step 3: Set up the inequality
To ensure that \( n \) is real, we need:
\[ 196 - m^2 \geq 0 \]
This can be rearranged to:
\[ m^2 \leq 196 \]

### Step 4: Solve for \( m \)
Taking the square root of both sides gives:
\[ |m| \leq 14 \]
This means:
\[ -14 \leq m \leq 14 \]

### Step 5: Find the largest value of \( m \)
Since we are looking for the largest value of \( m \), we take:
\[ m = 14 \]

### Conclusion
Thus, the largest value of \( m \) for which there exists a real value of \( n \) is:
\[ \boxed{14} \]

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