The mean proportional between `(3+sqrt(2))` and `(12 - sqrt(32))` is:

Options

A. `sqrt(7)`
B. `2sqrt(7)`
C. 6
D. `(15- 3sqrt(2))/2`

Answer

To find the mean proportional between \( (3 + \sqrt{2}) \) and \( (12 - \sqrt{32}) \), we will follow these steps:

### Step 1: Understand the formula for mean proportional
The mean proportional \( m \) between two numbers \( x \) and \( y \) can be calculated using the formula:
\[ m = \sqrt{x \cdot y} \]

### Step 2: Identify the values of \( x \) and \( y \)
Here, we have:
\[ x = 3 + \sqrt{2} \]
\[ y = 12 - \sqrt{32} \]

### Step 3: Simplify \( y \)
First, we simplify \( \sqrt{32} \):
\[ \sqrt{32} = \sqrt{16 \cdot 2} = 4\sqrt{2} \]
So, we can rewrite \( y \):
\[ y = 12 - 4\sqrt{2} \]

### Step 4: Calculate the product \( x \cdot y \)
Now we need to calculate \( x \cdot y \):
\[ x \cdot y = (3 + \sqrt{2})(12 - 4\sqrt{2}) \]
Using the distributive property (FOIL method):
\[ = 3 \cdot 12 + 3 \cdot (-4\sqrt{2}) + \sqrt{2} \cdot 12 + \sqrt{2} \cdot (-4\sqrt{2}) \]
\[ = 36 - 12\sqrt{2} + 12\sqrt{2} - 4 \cdot 2 \]
\[ = 36 - 8 \]
\[ = 28 \]

### Step 5: Find the mean proportional
Now, we can find the mean proportional:
\[ m = \sqrt{x \cdot y} = \sqrt{28} \]
We can simplify \( \sqrt{28} \):
\[ \sqrt{28} = \sqrt{4 \cdot 7} = 2\sqrt{7} \]

### Final Answer
Thus, the mean proportional between \( (3 + \sqrt{2}) \) and \( (12 - \sqrt{32}) \) is:
\[ \boxed{2\sqrt{7}} \]