Question

A local chicken farmer is constructing a set of eight pens to cage his chickens as shown in the diagram below. He has decided to use 1000 feet of fencing to create the pens.

(I'll attach diagram)

(a) Express the width y as a function of the length x.

y(x)=

(b) Express the total enclosed area A of the pens as function of x.

A(x)=


(c) Determine the dimensions x and y that will maximize the enclosed area.

x=________ feet
y=________ feet

Answer 1

Primary equation: A(x)= (5y)(3X)
Secondary equation: 5y+3X=1000
y=200-(3X)/5
A(x)=3X(1000-3X)
A(x)=3000X-9X²

Now, find the derivative of A(x) to find the max... here's the work for that, or you could guess and check.

A'(x)=3000-18X

Set derivative equal to 0

0=3000-18X
166.6666666=X

Now test the intervals

(0,166.6666) (166.66666, 1000)
1st derivative is + 1st derivative is -

Plug the X value back into the secondary equation
5y+3(166.666666)=1000
5Y=500
Y=5

Answer:
X= 166.6666666666
Y=5

Please note, this is entry level calculus, and your teacher may expect you to use a different, longer route such as guess and check.