Final answer:
The area of the parallelogram can be found by determining the base and height. The base is the horizontal distance between where the given lines intersect the constant function g(x)=2 and the height is the vertical distance from the x-axis to g(x)=2. The calculated area of the parallelogram is 34/3.
Step-by-step explanation:
The area of a parallelogram can be found by using the base and the height of the parallelogram. Given the parallelogram is bounded by the x-axis, the line g(x)=2, the line f(x)=3x, and a line parallel to f(x) passing through (6,1), we can calculate the area as follows:
- Identify the base of the parallelogram. Here, the base is the distance between the y-axes intersecting points of the line parallel to f(x) and the x-axis, which can be found as the x-coordinate of the point where this line intersects g(x)=2, and then subtract the x-coordinate of the point where f(x) intersects g(x)=2.
- To find the height, we look at the vertical distance between the x-axis and the line g(x)=2, which is simply 2.
- Multiply the base by the height to find the area of the parallelogram.
First, find the equation of the line parallel to f(x)=3x that passes through (6,1). Since the slope of f(x) is 3, the slope of the parallel line is also 3. This line has the equation y=3x+b. Plugging in the point (6,1), we get 1=3(6)+b, so b=-17, making the parallel line's equation y=3x-17.
To find the base, set the parallel line and g(x)=2 equal to each other: 3x-17=2, giving us x=19/3. The base extends from this intersection to where f(x)=3x meets g(x)=2, which is at (2/3, 2). Therefore, the length of the base is 19/3 - 2/3 = 17/3.
Now multiply the base (17/3) by the height (2) to get the area of the parallelogram: Area = (17/3) * 2 = 34/3.