The expression for the area of the triangle is Area = -0.5x² + 9x and the maximum possible area is 40.5 square units.
The formula for the area of a triangle is expressed as:
Area = 1/2 × base × height
From the diagram:
Height of the triangle = x
Base = 18 - x
Plug the height and the base of the triangle into the above formula and simplify:
Area = 1/2 × base × height
Area = 1/2 × ( 18 - x ) × x
Simplifying, we get:
Area = 1/2 × ( 18x - x² )
Area = -0.5x² + 9x
To determine the maximum possible area, firstly, we find the derivative, set it to zero, and solve for x:
f(x)= -0.5x² + 9x
f'(x)= -2( 0.5x ) + 9
f'(x)= -x + 9
Set f'(x) = 0:
-x + 9 = 0
x = 9
This is the critical point.
Now, we determine whether the point is maximum or minimum:
f''(x)= -2(0.5)
f''(x)= -1
Since the second derivative is negative, the critical point is a maximum.
Finally, plug x = 9 into the expression for the area and simplify:
f(x) = -0.5x² + 9x
f(9) = -0.5(9)² + 9(9)
f(9) = -0.5(81) + 81
f(9) = -40.5 + 81
f(9) = 40.5 square units.
Therefore, the maximum possible area is 40.5 square units.