Rabbit Cage

Question

Suppose you are enclosing a rectangular area to create a rabbit cage. You have 80ft of fence and want to build a pen with the largest possible area for your rabbit, so you build a cage using the existing side of a building as one side. 

 

A) Make a table showing the area for some selected values of x, and write a function that gives the area y, as function of width, x.

B) What width maximizes the area?

C) What is the maximum area for your rabbit?

Answer 1

you can use simple arithmetic to solve the problem or use algebra

the maximum area is always a square

80/4=20

you want a square 20 feet by 20 feet, but...

one side is the building

take the 20 feet that is not needed because of the building and add it to the opposite side to get 20+20=40 feet

you have a rectangle that is 20 by 40 by 20 by "the building"

A=l*w

A=20*40

A=800 square feet is the maximum area

notice that a 20 by 40 rectangle is the same as two 20 by 20 squares put together

using algebra...

P=2l+2w

80=2l+2w

40=l+w

40-w=l

A=l*w

A=(40-w)*(w)

A=40w-w²

use -b/2a where (h,k) is the vertex of the parabola (the parabola opens down-cap not cup) and h=-b/2a and k=maximum area

-b/2a=-40/-2=20 feet again and go from here

cap=∩ and cup=U