Question

Find the area of the largest rectangle in the first quadrant that would have a side on each of the axes and a vertex on the line y = -1/3x + 1.

Answer 1

The area of the largest rectangle in the first quadrant that would have a side on each of the axes and a vertex on the line y = -1/3x + 1 is 0 square units.

Step-by-step explanation:

To find the area of the largest rectangle in the first quadrant with a side on each of the axes and a vertex on the line y = -1/3x + 1, we need to determine the dimensions of the rectangle. The rectangle will have one vertex on the line, another vertex on the x-axis (where y = 0), and another vertex on the y-axis (where x = 0). Let's calculate the values:

1. The x-coordinate of the vertex on the line y = -1/3x + 1 is obtained by solving the equation -1/3x + 1 = 0 for x. This gives x = 3.
2. Therefore, the y-coordinate of the vertex on the line is y = -1/3(3) + 1 = 0.
3. The height of the rectangle is the y-coordinate of the vertex on the line, which is 0.
4. The width of the rectangle is the x-coordinate of the vertex on the line, which is 3.

Now, we can calculate the area of the rectangle using the formula A = length × width. In this case, the length is the height (0) and the width is 3. Therefore, the area of the largest rectangle is 0 square units.