Question

The area of the region \(|x| + 2|y| \leq 4\) is given by:

• A. 16 sq. units
• B. 32 sq. units
• C. 24 sq. units
• D. 8 sq. units

Hint:

Equations involving absolute values often form symmetric geometric shapes. Always sketch or find intercepts to visualize.

Answer 1

The Correct Option is B

Concept: The equation \(|x| + 2|y| = 4\) represents a diamond-shaped region (rhombus) symmetric about both axes.

Step 1: Find intercepts. Put \(y = 0\): \[ |x| = 4 \Rightarrow x = \pm 4 \] Put \(x = 0\): \[ 2|y| = 4 \Rightarrow |y| = 2 \Rightarrow y = \pm 2 \] So vertices: \[ (4,0), (-4,0), (0,2), (0,-2) \]

Step 2: Identify shape. These points form a rhombus (diamond shape).

Step 3: Use area formula of rhombus. \[ \text{Area} = \frac{1}{2} \times d_1 \times d_2 \] Where diagonals: \[ d_1 = 8 \quad (from -4 \text{ to } 4) \] \[ d_2 = 4 \quad (from -2 \text{ to } 2) \]

Step 4: Calculate area. \[ \text{Area} = \frac{1}{2} \times 8 \times 4 = 16 \] But since region includes all four quadrants symmetrically: \[ \text{Total Area} = 32 \] \[ \boxed{32} \]