A farmer plans to use 21 m of fencing to enclose a rectangular pen having an area of 55 m². Only three sides of the pen need fencing because of an existing wall. Find the dimensions of the pen.

Question

A farmer plans to use 21 m of fencing to enclose a rectangular pen having an area of 55 m².  Only three sides of the pen need fencing because of an existing wall.  Find the dimensions of the pen.

anonymous · Answer

For this problem you need to deal with both perimeter and area so let's start with those formulas

If the length is denoted by l and the width by w:
P = 2l + 2w
A = l

Now we only are concerned with three sides of the pen when dealing with perimeter so using the 21m of material he has, we see that
21 = l + 2w (since the other length is using the existing wall.

We want the area to be 55m^2 so plugging this in we have
55 = lw

Solving the perimeter equation for l and substituting, we get$$55=lw=(21-2w)w=21w-2w^{2}$$$$-2w^{2}+21w=55$$
We can now solve this quadratic by moving the 55 to the left and factoring
$$-2w^{2}+21w-55=0$$$$2w^{2}-21w+55=0$$(2w - 11)(w - 5)=0

Solving each of these factors for w will give us two posible values for the width which we can plug into the perimeter equation to get the associated lengths.

I leave the rest up to you...

anonymous · Answer

5 by 11 (by eye)

anonymous · Answer

That's only one of the answers @estudier.  Since it's a quadratic, there are 2 solutions.

anonymous · Answer

5.5 m by 10 m or 5 m by 11 m

anonymous · Answer

The other is probably something daft, like a negative (guessing)

anonymous · Answer

thats what i got

anonymous · Answer

What's the max area you can enclose?

anonymous · Answer

@nany - You are correct!  Outstanding work!

@estudier - No, the other solution is not daft.  Many of these area/perimeter problems have to viable solutions.  One is usually nice numbers like 5 x 11 and the other is less nice because you're reallocating "blocks" from the side and redistribuiing them to the length.

Perhaps, @estudier, you should work out the problem using the corret method prior to assuming answers and giving false information.

anonymous · Answer

Thank You So Much For Your Help !!

anonymous · Answer

A = x(21-2x)  so dA = 21-4x = 0 for max A --> x=21/4 and max A = 55 and an eighth.

@mtbender74, I gave no false information  (My first answer I said was by eye and the second I said was a guess, I apologize if you misunderstood me).