A rectangular pen for pigs that will enclose a total area of 169 square feet. What is the least amount of fencing that is needed?

The answer is below

Step-by-step explanation:

Let x represent the length of the rectangular pen and y represent the width of the rectangular pen. Since the total area of the pen is 169 ft², hence:

Area = length * width

169 = xy

y = 169/x

Also the perimeter of the rectangular pen is:

Perimeter (P) = 2(length + width)

P = 2(x + y)

P = 2x + 2y

P = 2x + 2(169/x)

P = 2x + 338/x

The least amount of fencing is at dP/dx = 0, hence:

dP/dx = 2 - 338/x² = 0

338/x² = 2

2x² = 338

x² = 169

x = 13 feet

y = 169 / x = 169/13 = 13 feet

The least amount of fencing = P = 2(x + y) =  2(13 + 13) = 52 feet

Hey there! Hello!

Not sure if you still need these answers, but I'd love to help out if you do! 

Now, I want you to go ahead and think of some stuff that's true for squares. To name a few, the opposite sides are going to be parallel to one another, all the angles are 90°, all the sides are the same length, and both diagonals are going to be perpendicular and equal in length. I'm sure there's even more, but I'll leave that to you. (BTW, by diagonals, I mean the lines that go through the the opposite diagonal corners). 

What about rectangles? The opposite sides are going to be parallel to one another, the diagonals are going to be equal in length, and the angles are going to be 90°. 

Now, rhombi. All sides are going to be equal, opposite sides are going to be parallel, the diagonally opposite angles will be equal to each other, and the diagonals bisect each other at 90°. 

And lastly, parallelograms. Pretty similar to rhombi in that they have parallel opposite sides and that the opposite diagonal angles are equal to each other, but there's one thing that makes a parallelogram not a rhombus. 

If you differentiate the stuff I described, you'll be golden. There's a lot to choose from, and I personally like to have options. Hope this helped you out, feel free to ask me any additional questions you have! :-)