The area is given by A = -2x² + 120x. The greatest area is given by x = 30 m.
Explanation:
See the picture attached for reference.
Let's call:
x = side not facing the wall
We know that the total fence is 120m, therefore
(120 - 2x) = side facing the wall
Note: you could choose to be x = side facing the wall, but the calculations would be a little bit more complicated.
We can now calculate the area of a rectangle:
A = b · h =
= x · (120 - 2x)
= -2x² + 120x
In order to find a maximum for this function, we need to calculate the first derivative:
[tex] \frac{d}{dx} (-2x^{2} + 120x ) = -4x +120[/tex]
Then, we need to find the candidate points by setting the derivative equal to zero:
-4x +120 = 0
-4(x - 30) = 0
x = 30
Now, in order to understand if the candidate point is a maximum or a minium, let's calculate the second derivative:
[tex] \frac{d}{dx}(-4x + 120) = -4[/tex]
According to the "Second Derivative Test", if the second derivative is negative, the point is a local maximum.
Hence, x = 30m gives the greatest area, and we would have:
side not facing the wall = 30m
side facing the wall = 120 - 2·30 = 60m
area = 30 · 60 = 1800 m²
