Problem 21
Question
Use inequalities to describe \(R\) in terms of its vertical and horizontal cross sections.\(R\) is the rectangle with vertices \((-1,1),(2,1)\), \((2,2)\), and \((-1,2)\).
Step-by-Step Solution
Verified Answer
R is given by -1 ≤ x ≤ 2 and 1 ≤ y ≤ 2.
1Step 1: Identify the rectangle vertices
Determine the vertices of the rectangle. The given vertices are (-1,1), (2,1), (2,2), and (-1,2).
2Step 2: Determine the horizontal range
From the vertices, identify the minimum and maximum values of the x-coordinates. The minimum x-coordinate is -1 and the maximum x-coordinate is 2. Thus, the horizontal range can be described by the inequality -1 ≤ x ≤ 2.
3Step 3: Determine the vertical range
From the vertices, identify the minimum and maximum values of the y-coordinates. The minimum y-coordinate is 1 and the maximum y-coordinate is 2. Thus, the vertical range can be described by the inequality 1 ≤ y ≤ 2.
4Step 4: Combine the inequalities
The region R can be fully described by combining the horizontal and vertical inequalities. Thus, R is given by the following system of inequalities: -1 ≤ x ≤ 2 and 1 ≤ y ≤ 2.
Key Concepts
rectangle verticeshorizontal range inequalityvertical range inequalitysystem of inequalities
rectangle vertices
In geometry, the vertices of a rectangle are crucial. They define the shape and position of the rectangle on a coordinate plane.
For this exercise, we have the vertices at points ewline ewline ewline (-1,1), (2,1), (2,2), and (-1,2).
The vertices tell us where each corner of the rectangle is located.
To find these points, we look at where the lines intersect. This helps us understand the boundaries of the rectangle.
Once we have the vertices, we can explore further properties like the lengths of sides and range inequalities.
For this exercise, we have the vertices at points ewline ewline ewline (-1,1), (2,1), (2,2), and (-1,2).
The vertices tell us where each corner of the rectangle is located.
To find these points, we look at where the lines intersect. This helps us understand the boundaries of the rectangle.
Once we have the vertices, we can explore further properties like the lengths of sides and range inequalities.
horizontal range inequality
The horizontal range describes the span of the rectangle along the x-axis. For this, we need to determine the minimum and maximum x-coordinates of the vertices.
In our exercise, the x-coordinates of the vertices are -1 and 2.
This gives us a horizontal range from -1 to 2.
We can express this relationship mathematically using an inequality: ewline ewline ewline -1 ≤ x ≤ 2. ewline ewline ewline
This range tells us that the x-values for any point inside the rectangle must lie between -1 and 2.
In our exercise, the x-coordinates of the vertices are -1 and 2.
This gives us a horizontal range from -1 to 2.
We can express this relationship mathematically using an inequality: ewline ewline ewline -1 ≤ x ≤ 2. ewline ewline ewline
This range tells us that the x-values for any point inside the rectangle must lie between -1 and 2.
vertical range inequality
The vertical range describes the extent of the rectangle along the y-axis. To find this range, we examine the y-coordinates of the vertices.
For our rectangle, the y-coordinates are 1 and 2.
This gives us a vertical range from 1 to 2.
We use the following inequality to describe this range: 1 ≤ y ≤ 2.
This tells us that any point within the rectangle must have a y-value that falls between 1 and 2.
For our rectangle, the y-coordinates are 1 and 2.
This gives us a vertical range from 1 to 2.
We use the following inequality to describe this range: 1 ≤ y ≤ 2.
This tells us that any point within the rectangle must have a y-value that falls between 1 and 2.
system of inequalities
To completely describe the area covered by the rectangle, we need to combine the horizontal and vertical range inequalities.
These combined inequalities form a system that defines the rectangle.
For our given rectangle, the system of inequalities is: ewline ewline ewline -1 ≤ x ≤ 2 and 1 ≤ y ≤ 2. ewline ewline ewline
This system shows that any point (x,y) inside the rectangle must satisfy both the horizontal inequality (-1 ≤ x ≤ 2) and the vertical inequality (1 ≤ y ≤ 2).
Using this system, we can fully capture and describe the two-dimensional region occupied by the rectangle.
These combined inequalities form a system that defines the rectangle.
For our given rectangle, the system of inequalities is: ewline ewline ewline -1 ≤ x ≤ 2 and 1 ≤ y ≤ 2. ewline ewline ewline
This system shows that any point (x,y) inside the rectangle must satisfy both the horizontal inequality (-1 ≤ x ≤ 2) and the vertical inequality (1 ≤ y ≤ 2).
Using this system, we can fully capture and describe the two-dimensional region occupied by the rectangle.
Other exercises in this chapter
Problem 19
Use inequalities to describe \(R\) in terms of its vertical and horizontal cross sections.\(R\) is the region bounded by \(y=x^{2}\) and \(y=3 x\).
View solution Problem 20
Use inequalities to describe \(R\) in terms of its vertical and horizontal cross sections.\(R\) is the region bounded by \(y=\sqrt{x}\) and \(y=x^{2}\).
View solution Problem 22
Use inequalities to describe \(R\) in terms of its vertical and horizontal cross sections.\(R\) is the triangle with vertices \((1,0),(1,1)\), and \((2,0)\).
View solution Problem 23
Use inequalities to describe \(R\) in terms of its vertical and horizontal cross sections.\(R\) is the region bounded by \(y=\ln x, y=0\), and \(x=e\).
View solution