A solid rectangular block has a square base with side `(sqrt3 - sqrt2)` m. The volume of the block is `(3sqrt18 - 7sqrt3) m^(3)` . Find the height of the block in the form `(asqrt2 +bsqrt3)`m, where a and b are integers.

Options

A. `(3sqrt2 - sqrt3)` m
B. `(5sqrt3 + 4sqrt2)` m
C. `(15sqrt2 + sqrt3)` m
D. `(5sqrt5 + 3sqrt3)` m

Answer

To find the height of the solid rectangular block with a square base and given volume, we will follow these steps:

### Step 1: Understand the dimensions of the block
The block has a square base with side length \( x = \sqrt{3} - \sqrt{2} \) m. The height of the block is denoted as \( h \).

### Step 2: Write the formula for the volume of the block
The volume \( V \) of a rectangular block (cuboid) is given by the formula:
\[ V = \text{length} \times \text{breadth} \times \text{height} \]
For our block, since it has a square base:
\[ V = x^2 \cdot h \]

### Step 3: Substitute the values into the volume formula
Substituting \( x = \sqrt{3} - \sqrt{2} \):
\[ V = (\sqrt{3} - \sqrt{2})^2 \cdot h \]

### Step 4: Calculate \( (\sqrt{3} - \sqrt{2})^2 \)
Using the formula \( (a - b)^2 = a^2 - 2ab + b^2 \):
\[ (\sqrt{3} - \sqrt{2})^2 = 3 - 2\sqrt{3}\sqrt{2} + 2 = 5 - 2\sqrt{6} \]
Thus, we have:
\[ V = (5 - 2\sqrt{6}) \cdot h \]

### Step 5: Set the volume equal to the given volume
The volume of the block is given as \( V = 3\sqrt{18} - 7\sqrt{3} \). We can simplify \( 3\sqrt{18} \):
\[ 3\sqrt{18} = 3 \cdot 3\sqrt{2} = 9\sqrt{2} \]
So, the volume can be rewritten as:
\[ V = 9\sqrt{2} - 7\sqrt{3} \]

### Step 6: Equate the two expressions for volume
Now we set the two expressions for volume equal to each other:
\[ (5 - 2\sqrt{6}) \cdot h = 9\sqrt{2} - 7\sqrt{3} \]

### Step 7: Solve for \( h \)
To find \( h \), we rearrange the equation:
\[ h = \frac{9\sqrt{2} - 7\sqrt{3}}{5 - 2\sqrt{6}} \]

### Step 8: Rationalize the denominator
To simplify \( h \), we multiply the numerator and denominator by the conjugate of the denominator:
\[ h = \frac{(9\sqrt{2} - 7\sqrt{3})(5 + 2\sqrt{6})}{(5 - 2\sqrt{6})(5 + 2\sqrt{6})} \]
Calculating the denominator:
\[ (5 - 2\sqrt{6})(5 + 2\sqrt{6}) = 5^2 - (2\sqrt{6})^2 = 25 - 24 = 1 \]
So, the denominator simplifies to 1.

### Step 9: Expand the numerator
Now we expand the numerator:
\[ (9\sqrt{2} - 7\sqrt{3})(5 + 2\sqrt{6}) = 9\sqrt{2} \cdot 5 + 9\sqrt{2} \cdot 2\sqrt{6} - 7\sqrt{3} \cdot 5 - 7\sqrt{3} \cdot 2\sqrt{6} \]
Calculating each term:
\[ = 45\sqrt{2} + 18\sqrt{12} - 35\sqrt{3} - 14\sqrt{18} \]
Since \( \sqrt{12} = 2\sqrt{3} \) and \( \sqrt{18} = 3\sqrt{2} \):
\[ = 45\sqrt{2} + 36\sqrt{3} - 35\sqrt{3} - 42\sqrt{2} \]
Combining like terms:
\[ = (45\sqrt{2} - 42\sqrt{2}) + (36\sqrt{3} - 35\sqrt{3}) = 3\sqrt{2} + \sqrt{3} \]

### Final Result
Thus, the height \( h \) is:
\[ h = 3\sqrt{2} - \sqrt{3} \text{ m} \]