The maximum possible area of the rectangle is 12√3 square units.
How to find the maximum possible area ?
The area "A" of the rectangle is given by:
A = 2x * y
Find an expression for "y" in terms of "x" using the equation of the parabola:
y = 9 - x ²
To find the maximum area, maximize the area function A(x) = 2x * (9 - x ²) by taking its derivative and setting it equal to zero:
A'(x) = 2(9 - x ²) - 2x(2x) = 18 - 2x ² - 4x ² = 18 - 6x ²
Setting A'(x) equal to zero and solving for x:
18 - 6x ² = 0
6x ² = 18
x ² = 3
x = ±√3
Plug this value of x back into the equation for y:
y = 9 - 3
y = 6
So, the dimensions of the rectangle that maximize its area are a base of 2√3 and a height of 6. Now, calculate the maximum area:
A = 2x * y = 2(√3) * 6
= 12√3 square units
Therefore, the maximum possible area of the rectangle is 12√3 square units.