If `sqrt(3)=1.732`, then find the value of `4(sqrt(192)-(1)/(2)sqrt(48)-sqrt(75))` correct to 3 places of decimal.

Options

A. `6.928`
B. `5.928`
C. `2.732`
D. `3.732`

Answer

To solve the expression \( 4(\sqrt{192} - \frac{1}{2}\sqrt{48} - \sqrt{75}) \) using the given value of \( \sqrt{3} = 1.732 \), we will break it down step by step.

### Step 1: Simplify \( \sqrt{192} \)
We can express \( 192 \) as \( 16 \times 12 \).
\[ \sqrt{192} = \sqrt{16 \times 12} = \sqrt{16} \times \sqrt{12} = 4\sqrt{12} \]

### Step 2: Simplify \( \sqrt{48} \)
We can express \( 48 \) as \( 16 \times 3 \).
\[ \sqrt{48} = \sqrt{16 \times 3} = \sqrt{16} \times \sqrt{3} = 4\sqrt{3} \]

### Step 3: Simplify \( \sqrt{75} \)
We can express \( 75 \) as \( 25 \times 3 \).
\[ \sqrt{75} = \sqrt{25 \times 3} = \sqrt{25} \times \sqrt{3} = 5\sqrt{3} \]

### Step 4: Substitute back into the expression
Now we can substitute the simplified square roots back into the original expression:
\[ 4(\sqrt{192} - \frac{1}{2}\sqrt{48} - \sqrt{75}) = 4(4\sqrt{12} - \frac{1}{2}(4\sqrt{3}) - 5\sqrt{3}) \]

### Step 5: Simplify the expression inside the parentheses
\[ = 4(4\sqrt{12} - 2\sqrt{3} - 5\sqrt{3}) = 4(4\sqrt{12} - 7\sqrt{3}) \]

### Step 6: Calculate \( 4\sqrt{12} \)
We can express \( \sqrt{12} \) as \( 2\sqrt{3} \):
\[ 4\sqrt{12} = 4(2\sqrt{3}) = 8\sqrt{3} \]

### Step 7: Substitute back into the expression
\[ = 4(8\sqrt{3} - 7\sqrt{3}) = 4(1\sqrt{3}) = 4\sqrt{3} \]

### Step 8: Substitute the value of \( \sqrt{3} \)
Now we substitute the value of \( \sqrt{3} = 1.732 \):
\[ = 4 \times 1.732 = 6.928 \]

### Final Answer
Thus, the value of \( 4(\sqrt{192} - \frac{1}{2}\sqrt{48} - \sqrt{75}) \) correct to three decimal places is:
\[ \boxed{6.928} \]