the point P lies in the first quadrant on the line y=10-4x from P, perpendicular lines are drawn to the x and y axis. What is the maximum area of the rectangle formed?
To find the maximum area of the rectangle formed, we need to first determine the coordinates of point P in the first quadrant. The line equation given is y = 10 - 4x. Since point P lies on this line, we can substitute the values of x and y into the equation to find their relationship.
Let's assume the coordinates of point P are (a, b).
Substituting these values into the line equation, we have:
b = 10 - 4a
To find point P, we need another piece of information. The problem states that from point P, perpendicular lines are drawn to the x and y-axis. This means that the rectangle formed is a right-angled triangle.
Let's examine the triangle formed by the x-axis, y-axis, and the line y = 10 - 4x.
The x-coordinate of the point where the perpendicular line intersects the x-axis is a. The y-coordinate of the point where the perpendicular line intersects the y-axis is b.
The area of the rectangle formed is given by the product of the x-coordinate and y-coordinate of point P, which is ab.
To maximize the area of the rectangle, we need to find the maximum value of ab.
To do this, we can use the given equation b = 10 - 4a and rewrite it as a function of a, which is b(a) = 10 - 4a.
Next, we can find the derivative of this function with respect to a, and set it equal to 0 to find critical points:
b'(a) = -4
Setting b'(a) equal to 0, we find -4 = 0. However, this equation has no solution, which means there are no critical points.
Since there are no critical points, we can conclude that the function b(a) = 10 - 4a is always decreasing.
Therefore, to maximize the area of the rectangle (ab), we need to use the extreme values of a within the constraints of the problem.
Since point P lies in the first quadrant, both a and b must be positive. We also know that the line intersects with the x-axis at (2.5, 0), which means a must be between 0 and 2.5.
Substituting the extreme values into the equation b(a) = 10 - 4a, we have:
For a = 0: b(0) = 10 - 4(0) = 10
For a = 2.5: b(2.5) = 10 - 4(2.5) = 10 - 10 = 0
So, the maximum area of the rectangle formed is given by the product of the coordinates of point P, which is 2.5 * 10 = 25 square units.