Final answer:
To find the area of the triangle formed by the y-axis, the line f(x), and a line perpendicular to f(x) through the origin, we must find the point of intersection between f(x) and the perpendicular line, use it to determine the base and height of the triangle, and then apply the area formula for a right triangle.
Step-by-step explanation:
The area of a triangle bounded by the y-axis, the line f(x) = 7 - \(\frac{3}{5}x\), and a line perpendicular to f(x) that passes through the origin can be found using the concept of finding the area under a curve. In this case, it involves calculating the area of a right triangle formed by these lines. The line perpendicular to f(x) will have a slope that is the negative reciprocal of the slope of f(x), which is \(\frac{5}{3}\). Given that this line passes through the origin, its equation can be written as y = \(\frac{5}{3}\)x. To find where this line intersects with f(x), we can set the equations equal to each other and solve for x.
Setting f(x) equal to the perpendicular line equation:
7 - \(\frac{3}{5}x\) = \(\frac{5}{3}\)x, we find the x-coordinate of the intersection point. Once we have the coordinates of the point of intersection, we can calculate the base and the height of the right triangle. The base is the x-coordinate of the intersection point, and the height is the corresponding y-coordinate since the triangle is bounded by the y-axis and the lines. Finally, the area of the right triangle is calculated using the formula \(\frac{1}{2} \times \text{base} \times \text{height}\).