What are the dimensions of the largest area that can be enclosed?

Question

The owner of the Rancho Los Feliz has 3480 yd of fencing to enclose a rectangular piece of grazing land along the straight portion of a river and then subdivide it by means of a fence running parallel to the sides. No fencing is required along the river. What are the dimensions of the largest area that can be enclosed?

Answer 1

First draw a general form for your fence, then we will set the perimeter equal to 3480 yd.(accounting for the river) and finally optimize area as a function of a length somewhere(doesn't matter, it'll come out the same). If this comes with an image, use it. You can set the right sides to the river as a variable, L. the river and the side opposite the river will be W. Whether the dividing fence runs parallel or perpendicular to the river will determine its variable but I'll be assuming it's running right to the river, so it's length will be L As well. Thus your fencing is now 3"L"'s and 1 "W", which you can write as an expression 3L+W. This is your fencing, and you'll be using it all, waste not, want not. Thus you find

3L+W=3480        (1)

(I will use is and "=" interchangably). Now you have your hypothetical fence, it's nice. What is it's area? Area is the product of length and width so A=L•W                (2)

All well and good but you cannot derlive th area with two variables. So you need to write one variable in terms of the other. Solving for W gives W=3480-3L        (3)

from (1). Substituting W from (3) into (2) gives you

A=L•(3480-3L)    (4)

Now there's two methods to solve this depending on if you know calculus. You can calculus through, derive and find your zero and test if this is a maximum, or use algebra to find the vertex since (4) is quadratic. I'll show both

Calculus

deriving A from (4) gives

dA=3480-6L        (5)

Since you only need the extrema of this dA=0 and L=3480/6

 

Algebra

Multiplying out (4) gives

A=-3L^2+3480L

The vertex is the highest point and

-b/(2a) so L is 3480/6 and W is 3480/2