What is the area of the quadrilateral bounded by #y=5#; #x=1#; #y=1#; and #y=2x+9#?
Note that
making it easier to evaluate the intersection coordinates for
From the above diagram it is clear that the area is
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To find the area of the quadrilateral bounded by the given equations, we need to first identify the vertices of the quadrilateral by finding their intersection points. Then, we can use the formula for finding the area of a quadrilateral formed by four given points.

Find the intersection points of the given equations:
 For y = 5 and x = 1: The intersection point is (1, 5).
 For y = 1 and x = 1: The intersection point is (1, 1).
 For y = 2x + 9 and y = 1: Substitute y = 1 into y = 2x + 9: 1 = 2x + 9 Solving for x: x = 4. Substitute x = 4 into y = 2x + 9: y = 1. The intersection point is (4, 1).
 For y = 2x + 9 and y = 5: Substitute y = 5 into y = 2x + 9: 5 = 2x + 9 Solving for x: x = 2. Substitute x = 2 into y = 2x + 9: y = 5. The intersection point is (2, 5).

Now, we have the vertices of the quadrilateral: (1, 5), (1, 1), (4, 1), and (2, 5).

Calculate the area of the quadrilateral using these vertices. We can divide the quadrilateral into two triangles and find the area of each triangle using the formula for the area of a triangle (0.5 * base * height), then sum the areas of the triangles.
Triangle 1: Vertices (1, 5), (1, 1), and (4, 1).
 Base = Distance between (1, 5) and (4, 1) = √((41)^2 + (15)^2) = √(3^2 + (4)^2) = √(9 + 16) = √25 = 5 units
 Height = Distance between (1, 1) and (1, 5) = 5  1 = 4 units
 Area = 0.5 * base * height = 0.5 * 5 * 4 = 10 square units
Triangle 2: Vertices (1, 1), (4, 1), and (2, 5).
 Base = Distance between (1, 1) and (2, 5) = √((21)^2 + (51)^2) = √(1^2 + 4^2) = √(1 + 16) = √17 units
 Height = Distance between (1, 1) and (4, 1) = 4  1 = 3 units
 Area = 0.5 * base * height = 0.5 * √17 * 3 = 1.5√17 square units
 Total area of the quadrilateral = Area of Triangle 1 + Area of Triangle 2 = 10 + 1.5√17 square units.
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When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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