applications to calculus questions A farmer has 80m length of fencing. He wants to use it to form 3 sides of a rectangular enclosure against an existing fence, which provides the 4th side. find the maximum area that he can enclose and give its dimensions. I know I have to use the principle of stationary points but I don't know how to begin the problem. 
 A: Let the two side lengths be $l,w$.
Then, $2 l + w = 80$ and you want to maximize $A=lw = l ( 80 - 2l)$.
Solve $\frac{dA}{dl}=0$ for $l$, see that $\frac{dA}{dl}$ changes sign so that this value of $l$ is a maximizer then plug it into the expression for $A$ to find the maximized area. 
Alternatively, note that $A$ specifies a parabola and look for its vertex as the maximizer since it opens downwards. 
A: I know I have to use the principle of stationary points but I don't know how to begin the problem.
The guidelines below (from Larson’s book) show you how to start (this and related problems).


*

*Identify all given quantities and all quantities to be determined. If possible, make a sketch and label it with any relevant measurements.
Here is the rectangular enclosure:

Given quantity: the length of fencing which is $80$
Quantities to be determined: The area $A$ of the rectangular enclosure and the dimensions $x$ and $y$.


*Write a primary equation for the quantity that is to be maximized or
minimized.
The quantity that is to be maximized is the area $A$. So, a primary equation is
$$A=xy$$


*Reduce the primary equation to one having a single independent variable.
This may involve the use of secondary equations relating the independent
variables of the primary equation.
From the step 1 we get the secondary equation $2x+y=80$ which implies
$$y=80-2x$$
Substituting this in the equation of step 2 we get
$$A(x)=x(80-2x)$$
$$A(x)=-2x^2+80x$$


*Determine the feasible domain of the primary equation. That is, determine
the values for which the stated problem makes sense.
$$0<x<40,\qquad 0<y<80$$


*Determine the desired maximum or minimum value by the calculus
techniques.
Now, we have to do the calculation. The first thing is to solve $A'(x)=0$ with respect the domain specified in the step 4.
