Answer:
The dimensions of the field to maximise the area are 100 metres (width) and 200 metres (height).
Step-by-step explanation:
The formulas for the area (), measured in square metres, and the perimeter (), measured in metres, of the rectangle are, respectively:
(1)
(2)
Where:
- Width, measured in metres.
- Height, measured in metres.
Note: We assume that height of the rectangle is parallel to the wall of the house.
By (2):
In (1):
(3)
Then, we obtain its first and second derivatives by Differential Calculus:
(4)
(5)
By equalising (4) to zero, we find the following critical value for :
And besides the Second Derivative Test, this solution is associated to an absolute maximum. Given that , then the maximum area enclosed by fencing is:
And the height of the triangle is:
The dimensions of the field to maximise the area are 100 metres (width) and 200 metres (height).