An ostritch farmer wants to enclose a rectangluar area

Question

An ostrich farmer wants to enclose a rectangular area and then divide it into six pens with fencing parallel to one side of the rectangle (see the figure below). There are 970 feet of fencing available to complete the job. What is the largest possible total area of the six pens?

The answer I've come up with is 14787/5 ft^2

My L=227.5

My W=65

It says my answer is wrong.

Many thanks if you can help!

Answer 1

6 pens require 7 short lengths, each @ x ft.

To enclose the pens requires 2 long lengths, each @ y ft.

Total length of fence = 970 ft = 7x + 2y

Total Enclosed Area = A = xy sq ft

Express Area as a function of one variable:

y = (970 - 7x) / 2 = 485 - 3.5x (ft)

A = xy = f(x) = x(485 - 3.5x) = - 3.5x^2 + 485x (sq ft)

Determine the maximum area via the calculus:

1. dA/dx = - 7x + 485 (set first derivative = 0 to determine extremum)

2. d²A/dx² = - 7 (the negative second derivative tells us the extremum is a maximum value, for all x)

Optimum x = 69.286 ft

Optimum y = 485 - 3.5x = 242.5 ft

Maximum A = 16,801.79 sq ft (ANS)

I hope this helps,

Gary

Answer 2

Draw and label and diagram!

w represents the "sides" and the 5 separators.

(970 - 7w) / 2 represents the length.

A = w(970 - 7w) / 2

A = (970w - 7w²) / 2

A = -3.5w² + 485w

Maximum at w = -485 / (2)(-3.5), or 69.2857