A rectangular field is built next to a house and is enclosed by 400 m of fencing for the 3 remaining sides. Determine the dimensions of the field to maximize the area to nearest tenth of a metre.

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).

15.36 per mile

but forgot to round

i am pretty sure