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In the xy-plane, what is the area of the region bounded by y +2x ≥ 3, y –x ≥ -6 and the line, that is perpendicular to x = 0 and passes
through the origin?
through the origin?
- a)9/4
- b)27/4
- c)9
- d)27/2
- e)Cannot be determined
Correct answer is option 'B'. Can you explain this answer?
Verified Answer
In the xy-plane, what is the area of the region bounded by y +2x ≥ ...
Given
- y +2x ≥ 3
- y –x ≥ -6
- Line perpendicular to x=0 and passing through origin
- The equation x = 0 represents the y-axis
- So, the given line is perpendicular to the y-axis and passes through the origin
- Therefore, the given line is the x-axis
To Find: Area bounded by the region y +2x ≥ 3, y –x ≥ -6 and x-axis
Approach:
1. For finding the area bounded by the region, we need to first draw the line segments y +2x = 3 and y – x = 6
2. Once we draw these line segment, we need to find the side of each line segment where the region specified in the question statement lies
2. Once we draw these line segment, we need to find the side of each line segment where the region specified in the question statement lies
- For finding the region, we will put the coordinates of the origin (0,0) in the inequality. If the inequality is satisfied, the region lies towards the side of the line containing the origin, else it lies on the opposite side of the line containing the origin.
3. Once we have the region, we will find the area of the region using the standard geometry formulas
Working out:
1. The line segment y +2x = 3 will intersect the y-axis at (3,0) and x-axis at 
- The region y +2x – 3 ≥ 0 will be satisfied by the region, which does not contain the origin, as putting (0,0) in the inequality does not satisfy the inequality.
2. Similarly, the line segment y-x ≥ 6 will intersect the y-axis at (-6,0) and x-axis at (6,0).
- The region y-x ≥ -6 will consist of the region, which contains the origin.
3. So, the vertex points of the region are ,(6,0) and (3, -3)
,(6,0) and (3, -3)
4. We know that area of a triangle = ½ * base * height
- Let’s consider the base as the x-axis, thus length of the base
- Height of the triangle would be the magnitude of y-coordinate of point (3, -3) = 3
5. Thus area of the triangle = 
Hence the correct answer is Option B.
Most Upvoted Answer
In the xy-plane, what is the area of the region bounded by y +2x ≥ ...
To find the area of the region bounded by the equation y = 2x, we need to determine the limits of integration.
First, let's solve the equation for x in terms of y:
y = 2x
x = y/2
The region is bounded by y, so the limits of integration for y will be the values where the curve intersects the y-axis. When x = 0, y = 2(0) = 0. So the lower limit of integration is 0.
To find the upper limit of integration, we need to determine the y-value where the curve intersects the x-axis. When y = 0, x = 0/2 = 0. So the upper limit of integration is also 0.
Therefore, the area of the region bounded by y = 2x is given by the integral:
A = ∫[0,0] 2x dx
Since the limits of integration are the same, the integral evaluates to zero. Thus, the area of the region bounded by y = 2x is 0.
First, let's solve the equation for x in terms of y:
y = 2x
x = y/2
The region is bounded by y, so the limits of integration for y will be the values where the curve intersects the y-axis. When x = 0, y = 2(0) = 0. So the lower limit of integration is 0.
To find the upper limit of integration, we need to determine the y-value where the curve intersects the x-axis. When y = 0, x = 0/2 = 0. So the upper limit of integration is also 0.
Therefore, the area of the region bounded by y = 2x is given by the integral:
A = ∫[0,0] 2x dx
Since the limits of integration are the same, the integral evaluates to zero. Thus, the area of the region bounded by y = 2x is 0.
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In the xy-plane, what is the area of the region bounded by y +2x ≥ 3, y –x ≥ -6 and the line, that is perpendicular to x = 0 and passesthrough the origin?a)9/4b)27/4c)9d)27/2e)Cannot be determinedCorrect answer is option 'B'. Can you explain this answer?
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