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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.

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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.

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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).

  1. 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:

enter image description here

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 $y$.

  1. 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 $$A=xy$$

  1. 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 $$A(x)=x(80-2x)$$ $$A(x)=-2x^2+80x$$

  1. Determine the feasible domain of the primary equation. That is, determine the values for which the stated problem makes sense.

$$0<x<40,\qquad 0<y<80$$

  1. 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.

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