Question 320471: I decide to take up alpaca farming. I will build an alpaca pen next to my house. (Since it is next to my house, I don't need any fencing on the side where my house is. My house is huge, so I dont have to worry about that side being too short.) I have 800 feet of fencing. I am going to make a rectangular fence.
a. if I want the pen to have an area of 15,200 sq feet, what should the dimensions be?
b. I want to give my alpacas as much room as possible. What is the most area I can have in this rectangular pen, using my 800 feet of fencing?
Answer by checkley77(12844)   (Show Source):
You can put this solution on YOUR website! YOUR RECTANGLAR AREA WILL HAVE 2 EQUAL SIDES & 1 LENGTH.
P=2W+L
800=2W+L
L=800-2W
A=LW
15,200=(800-2W)W
15,200=800W-2W^2
2W^2-800W+15,200=0
2(W^2-400W+7,600)=0
2(W-380)(W-20)=0
W-380=0
W=380 FT.
L=800-2*380=800-760=40 FT.
OR:
W-20=0
W=20 FT.
------------------------
L=800-2W
AREA=LW
(800-2W)*W
800W-2W^2
-2W^2+800W
-2(W^2-400W) NOW COMPLETE THE SQUARE.
(400/2)^2=200^2=40,000
-2(W^2-400W+40,000)=40,000
-2(W-200)^2=40,000
W-200=0
W=200 ANS. FOR EACH OF THE SHORT WIDTHS.
L=800-2*200
L=800-400=400 ANS. FOR THE LENGTH,
PROOF:
A=400*200=80,000 FT^2 IS THE MAX. AREA.
TEST:
USE 199 & 402 FT.
AREA=199*402=79,998 FT^2
TEST 201 & 398
AREA=201*398=79,998 FT^2
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