let x and y be real numbers. find the area enclosed by the graph of the following equation: abs(2x)+abs(2y)=1
To find the area enclosed by the graph of the equation |2x| + |2y| = 1, we can solve this equation step by step.
Step 1: Graph the equation
Start by graphing |2x| + |2y| = 1 on a coordinate grid. This equation describes a diamond-shaped region.
Step 2: Analyze the absolute value expressions separately
Consider the four cases based on the absolute value expressions.
Case 1: 2x ≥ 0, 2y ≥ 0
In this case, both absolute value expressions are positive. The equation simplifies to:
2x + 2y = 1
Solving for y, we get:
y = 0.5 - x/2
This is a line passing through the points (0,0.5) and (0.5,0) in the first quadrant.
Case 2: 2x ≥ 0, 2y < 0
In this case, the first absolute value expression is positive, but the second is negative. The equation simplifies to:
2x - 2y = 1
Rearranging, we get:
y = x/2 - 0.5
This is a line passing through the points (0,-0.5) and (1,0) in the fourth quadrant.
Case 3: 2x < 0, 2y ≥ 0
Here, the first absolute value expression is negative, while the second is positive. The equation simplifies to:
-2x + 2y = 1
Rearranging, we obtain:
y = x/2 + 0.5
This is a line passing through the points (0,0.5) and (-1,0) in the second quadrant.
Case 4: 2x < 0, 2y < 0
In this case, both absolute value expressions are negative. The equation simplifies to:
-2x - 2y = 1
Solving for y, we get:
y = -0.5 - x/2
This is a line passing through the points (0,-0.5) and (-0.5,0) in the third quadrant.
Step 3: Determine the region enclosed
The region enclosed by the graph can be obtained by considering the intersections of the lines in the four cases. The diamond-shaped region is formed by the four lines and is bounded by four points: (0,0.5), (1,0), (0,-0.5), and (-1,0).
Step 4: Calculate the area of the region
To find the area of this diamond-shaped region, we can calculate the area of two triangles and add them together.
Triangle 1: Base = 1, Height = 0.5
Area of Triangle 1 = (1 * 0.5) / 2 = 0.25
Triangle 2: Base = 1, Height = 0.5
Area of Triangle 2 = (1 * 0.5) / 2 = 0.25
Total Area = Area of Triangle 1 + Area of Triangle 2 = 0.25 + 0.25 = 0.5
Therefore, the area enclosed by the graph of |2x| + |2y| = 1 is 0.5 square units.