Quadratic Word Problem

Question

Mary is building a rectangular pen around her house for her little puppies. She subdivides it into 6 smaller rectangular pens (assume they are congruent). She plans to use the back wall of the house as one side of the pen. If she has 1400 feet of fencing to work with, what dimensions should she make each of the 6 smaller pens to maximize the area for her puppies to play in.

Answer 1

w is the width of the pen

L is the Length of the pen

The pen looks something like this:

back wall of house

--------------------------

w | | | | | | |

__________________________

so there are 7 widths and 2 only 1 length

7w + L = 1400

Area = L * w

= (1400 - 7w)*w

= 1400w - 7w^2

= A(w)

this is a quadratic function for Area

in terms of the width w

A=-7 B=1400, C = 0

max of the quadratic parabola is

-B/(2a) = -1400/ (2* -7)

= -1400 / -14

= 100

max occurs at w=100

The area is A(100) = (1400-7*100)*100

= (1400 -700)*100

= 700*100

= 70000

Length is area/width = 70000/100 = 700

the length of each sub-pen is 700/6 = 350/3 = 116 + 2/3

= 116 feet+ 8 inches

because 2/3 of 1 foot = 2/3*12 = 8 inches