Find the area of the region bounded by the graphs of y = x, y = −x + 4, and y = 0.
1
2
4
None of these
Make a sketch.
No calculus needed, we have a triangle with straight lines
Using some mental arithmetic and looking at my diagram, the base of my triangle is 4 units, and the height is 2
area = (1/2)(4)(2) = 4
To find the area of the region bounded by the given graphs, we need to find the points of intersection between the curves.
First, let's find the intersection points between y = x and y = -x + 4.
Setting the two equations equal to each other, we have:
x = -x + 4
Simplifying the equation, we get:
2x = 4
Dividing both sides by 2, we find:
x = 2
Substituting this value of x back into either equation, we find the corresponding y-value:
y = x = 2
Therefore, one of the intersection points is (2, 2).
Now, let's find the intersection points between y = -x + 4 and y = 0.
Setting y = -x + 4 to 0, we have:
-x + 4 = 0
Adding x to both sides, we get:
4 = x
Substituting this value of x back into either equation, we find the corresponding y-value:
y = -x + 4 = -4 + 4 = 0
Therefore, the other intersection point is (4, 0).
Now that we have the intersection points, we can visualize the region bounded by the graphs.
The region is a triangle with vertices (0, 0), (2, 2), and (4, 0).
To find the area of a triangle, we can use the formula:
Area = (base * height) / 2.
In this case, the base of the triangle is the distance between (0, 0) and (4, 0), which is 4 - 0 = 4 units.
The height of the triangle is the distance between the base and the vertex (2, 2), which is 2 units.
Plugging these values into the formula:
Area = (4 * 2) / 2 = 8 / 2 = 4 square units.
Therefore, the area of the region bounded by the graphs is 4 square units.
Hence, the correct answer is option 3: 4.
To find the area of the region bounded by the given graphs, we need to identify the points where these graphs intersect. Let's equate pairs of equations to find these points of intersection:
1. Setting y = x and y = -x + 4 equal to each other:
x = -x + 4
2x = 4
x = 2
By substituting this value back into either equation, we can find the corresponding y-coordinate. Using y = x:
y = 2
Therefore, the first point of intersection is (2, 2).
2. Setting y = x and y = 0 equal to each other:
x = 0
Here, we find that the second point of intersection is (0, 0).
3. Setting y = -x + 4 and y = 0 equal to each other:
-x + 4 = 0
x = 4
By substituting this value back into the second equation, we find the y-coordinate:
y = -4 + 4
y = 0
Therefore, the third point of intersection is (4, 0).
Now that we have identified the points of intersection, we can calculate the area of the region bounded by these graphs. We can see that this region forms a triangle, and the base of the triangle is the line segment connecting the points (0, 0) and (4, 0), which lies along the x-axis. The height of the triangle is the vertical distance from the x-axis to the point (2, 2).
The formula for calculating the area of a triangle is:
Area = (base * height) / 2
In this case, the base is 4 units (the difference in x-coordinates between the points (0, 0) and (4, 0)), and the height is 2 units (the difference in y-coordinates between the points (2, 2) and (2, 0)).
Plugging these values into the formula, we get:
Area = (4 * 2) / 2
Area = 8 / 2
Area = 4
Therefore, the area of the region bounded by the given graphs is 4.
So, the correct answer is 4.