A table screen has a 12-inch diagonal. If the length of the screen is `sqrt3` times longer than the width, what is the area, in square inches, of the screen ?

Options

A. 12
B. `12sqrt3`
C. 36
D. `36 sqrt3`

Answer

To solve the problem step by step, we will follow the information given in the question and use the Pythagorean theorem and the formula for the area of a rectangle.

### Step 1: Define Variables
Let the width of the screen be \( W \) inches. According to the problem, the length \( L \) of the screen is \( \sqrt{3} \) times the width:
\[ L = \sqrt{3} \cdot W \]

### Step 2: Use the Pythagorean Theorem
Since the diagonal of the rectangle is given as 12 inches, we can apply the Pythagorean theorem. In a rectangle, the relationship between the length, width, and diagonal can be expressed as:
\[ L^2 + W^2 = D^2 \]
where \( D \) is the diagonal. Substituting \( D = 12 \):
\[ L^2 + W^2 = 12^2 \]
\[ L^2 + W^2 = 144 \]

### Step 3: Substitute Length in Terms of Width
Now, substitute \( L \) with \( \sqrt{3} \cdot W \):
\[ (\sqrt{3} \cdot W)^2 + W^2 = 144 \]
Calculating \( (\sqrt{3} \cdot W)^2 \):
\[ 3W^2 + W^2 = 144 \]
Combine like terms:
\[ 4W^2 = 144 \]

### Step 4: Solve for Width
Now, divide both sides by 4:
\[ W^2 = \frac{144}{4} = 36 \]
Taking the square root of both sides gives:
\[ W = \sqrt{36} = 6 \]

### Step 5: Find Length
Now that we have the width, we can find the length:
\[ L = \sqrt{3} \cdot W = \sqrt{3} \cdot 6 = 6\sqrt{3} \]

### Step 6: Calculate the Area
The area \( A \) of the rectangle is given by:
\[ A = L \cdot W \]
Substituting the values of \( L \) and \( W \):
\[ A = (6\sqrt{3}) \cdot 6 = 36\sqrt{3} \]

### Final Answer
Thus, the area of the screen is:
\[ \boxed{36\sqrt{3}} \text{ square inches} \]