Find the area of region bounded by the y-axis, the line y=6, and the line y=1/2x.
Hint(s):
Try graphing the y-axis along with y=1/2x and y =6. What shape is formed by these three graphs?
To find the area of the region bounded by the y-axis, the line y=6, and the line y=1/2x, we need to determine the shape formed by these three graphs.
First, let's graph y=1/2x. This is a linear equation with a slope of 1/2 and y-intercept at the origin (0,0). We can plot a few points to sketch the line:
x = -2 -> y = 1/2(-2) = -1
x = -1 -> y = 1/2(-1) = -1/2
x = 0 -> y = 1/2(0) = 0
x = 1 -> y = 1/2(1) = 1/2
x = 2 -> y = 1/2(2) = 1
Connecting these points, we get a line that passes through the origin and has a positive slope. Let's label this line as L1.
Next, let's graph the line y=6. This is a horizontal line that passes through the point (0, 6).
Finally, we have the y-axis, which is a vertical line passing through the origin.
When we plot these three lines, we can see that they form a triangle. The base of the triangle is the line y=6, and the height is the intersection point of y=1/2x and y=6.
To find the intersection point, we set y=1/2x equal to y=6:
1/2x = 6
x = 12
Therefore, the intersection point is (12, 6).
To find the area, we need to calculate the base and the height of the triangle.
The base is simply the length of the line y=6, which is 12 units.
The height is the distance from the y-axis to the intersection point (12, 6). Since the y-axis is the vertical line passing through the origin, the distance is the y-coordinate of the intersection point, which is 6 units.
Now, we can plug in the values for base and height into the formula for the area of a triangle:
Area = 1/2 * base * height
= 1/2 * 12 * 6
= 36 square units
Therefore, the area of the region bounded by the y-axis, the line y=6, and the line y=1/2x is 36 square units.