"Published in Newark, California, USA"
Find the area of the largest rectangle having one side on the x-axis and inscribed in the triangle formed by the lines y = x , y = 0, and 3x + y = 20.
Solution:
To illustrate the problem, it is better to draw the figure and sketch the three given lines in Rectangular Coordinate System as follows
 |
| Photo by Math Principles in Everyday Life |
The first thing that we have to do is to find the length of a rectangle which is x and then find the width of a rectangle which is y.
To get the value of x for a rectangle, we need to use the lines y = x and 3x + y = 20.
The line 3x + y = 20 can be written as
Hence, the length of a rectangle is
To get the value of y for a rectangle, we need to use the lines y = x and y = 0.
Hence, the width of a rectangle is
We know that the area of a rectangle is
Next, we need to eliminate x at the above equation. We can use one of the three lines to substitute the value of x. The best is y = x because it is the simplest equation of the three lines. Substitute x = y to the above equation, we have
Take the derivative on both sides of the equation with respect to y, we have
Set dA/dy = 0 because we want to maximize the area of a rectangle.
Therefore,
Finally, the area of a rectangle is
