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Find the area of the region defined by the inequalities y ≥ x, y > -x + 1, and y < 10.
Published on October 26, 2024
Geometry
The area of the region defined by the inequalities y ≥ x, y > -x + 1, and y < 10 is 90.25 square units.
Question
area of y\ge \:x,\:y>\:-x+1,\:y<\:10
Answer
Let’s break down how to find the area of the region defined by the inequalities:
1. Understanding the Inequalities
- y ≥ x: This represents all the points on or above the line y = x.
- y > -x + 1: This represents all the points above the line y = -x + 1 (notice it’s strictly greater than, so the line itself is not included).
- y < 10: This represents all the points below the horizontal line y = 10.
2. Sketching the Region
To visualize the area, follow these steps:
- Draw the line y = x. Shade the area above this line.
- Draw the line y = -x + 1. Shade the area above this line (don’t include the line itself).
- Draw the line y = 10. Shade the area below this line.
- The region where all three shaded areas overlap is the area we want to find.
3. Finding the Vertices
The overlapping region will be a triangle. To calculate its area, we need the coordinates of its vertices. We find these by solving the systems of equations formed by the lines:
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Intersection of y = x and y = -x + 1:
Solving this system, we get x = 1/2, y = 1/2. So, one vertex is (1/2, 1/2). -
Intersection of y = x and y = 10:
Solving this, we get x = 10, y = 10. So, another vertex is (10, 10). -
Intersection of y = -x + 1 and y = 10:
Solving this, we get x = -9, y = 10. So, the last vertex is (-9, 10).
4. Calculating the Area
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Base and Height: We can use the segment between (-9, 10) and (10, 10) as the base of our triangle. This base has a length of 19 units. The height is the perpendicular distance from the point (1/2, 1/2) to the base, which is 9.5 units.
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Area Formula: The area of a triangle is (1/2) * base * height.
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Calculation: Area = (1/2) * 19 * 9.5 = 90.25 square units.
Therefore, the area of the region defined by the inequalities is 90.25 square units.