Question 816302: You have decided to become an alpaca farmer and need to make a rectangular pen for your animals to stay in. You have purchased 72 feet of fencing materials to make their pen, and you would like to maximize the area of the pen so the alpaca can be as comfortable as possible.
A) Use what you know about the perimeter to determine the dimensions of the pen in terms of x.
http://imageshack.us/photo/my-images/543/896r.png/
B) Use Part A to write an equation to model the possible area of the pen in terms of x .
C) Use the equation you wrote in Part B to determine the maximum possible area of the alpaca pen.
D) An alpaca will feel comfortable in a pen with about 30 ft^2 per alpaca. How many alpacas can fit comfortably in the pen you created ?
Thanks for any help.
Answer by TimothyLamb(4379)   (Show Source):
You can put this solution on YOUR website! 72 = 2L + 2w
a = Lw
L = a/w
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72 = 2L + 2w
72 = 2a/w + 2w
2a/w + 2ww/w - 72 = 0
2a/w + 2ww/w - 72w/w = 0
2a + 2ww - 72w = 0w
2a = 0 - 2ww + 72w
a(w) = -(2/2)w^2 + (72/2)w
a(w) = -w^2 + 36w
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the above quadratic equation is in standard form, with a=-1, b=36, and c=0
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to solve the quadratic equation, plug this:
-1 36 0
into this: https://sooeet.com/math/quadratic-equation-solver.php
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the quadratic vertex is a maximum at: ( w= 18, a(w)= 324 )
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Answer:
w = 18 ft
L = 18 ft
a(w)= 324 sq.ft
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Number of alpacas that fit comfortably in the pen: 324/30 = 10 alpacas
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