Answer to September 2, 2002 Problem

Don't Fence Me In!

 
A rancher in Colorado has 750 feet of fencing. He wants to enclose a rectangular region and then divide it into four pens with fencing parallel to one side of the rectangle (see diagram below).

What is the largest possible total area of the four pens?

 

Solution to Problem:

The answer is 14,062.5 square feet.

You can solve the problem with calculus or with precalculus. I will outline the precalculus solution:

Let w = the total width of the four pens.
Let x = the five lengths needed to enclose the four pens.


Then you can form two equations:
The total fencing is: 5x + 2w = 750
The Area is:         A = xw

Solve the first equation for w:
w = 375 - 2.5x

Now substitute this expression for w in the Area equation:
A = x (375 - 2.5x)

Simplifying,
A = 375x - 2.5 x^2,
which is the equation of a parabola opening downward. The greatest area would be at the vertex of the parabola.

Perhaps, it is easier to see this if you replace A by y.
Then it will be in terms of x and y like you are accustomed to seeing. Just remember y stands for the area:

y = 375x - 2.5 x^2

You can solve this on a graphing calculator. If you do, be sure to change the domain and range!!! (Use something like 50 to 100 for x; and use 12000 to 15000 for y).

You can also solve this by completing the square. This will give you the coordinates of the vertex (x, y), which will give you the length of each pen (x) and the area of the four pens (y), which, of course, is what you are looking for.

y = 375x - 2.5x^2

I would rewrite it as:
2.5 x^2 - 375 x = -y

2.5 (x^2 - 150x ) = -y

2.5 (x^2 - 150x + 5625) = -y + 2.5(5625)

2.5 (x^2 - 150x + 5625) = -y + 14,062.5

2.5 (x - 75)^2 = - (y - 14,062.5)

-2.5 (x - 75)^2 = y - 14,062.5

so the vertex is (75, 14062.5)
or in other words, the largest area is 14,062.5 square feet
when x = 75 feet. By the way, the other dimension
is 187.5 feet (this is the total width of the four pens).



Correctly solved by:

1. Richard K. Johnson La Jolla, California
2. Garrett Jue Memphis, Tennessee
3. George Gaither Winchester, Virginia
4. Peggah Sadeghzadeh Winchester, Virginia
5. Keith Mealy Cincinnati, Ohio
6. Tina Zahel Winchester, Virginia
7. Josh Feingold Winchester, Virginia
8. Walt Arrison Philadelphia, Pennsylvania
9. Izzy Kushner Closter, New Jersey
10. David and Judy Dixon Bennettsville, South Carolina
11. Matt Stewart Columbus, Georgia
12. Tiffany Abdullahi Memphis, Tennessee
13. Kathleen Altemose Winchester, Virginia
14. Jeffrey Gaither Winchester, Virginia
15. Matt McMurtry Arlington, Virginia
16. Rodolfo Hernandez Perez Reynosa Tamaulipas, Mexico
17. David Powell College of William and Mary, Williamsburg, Virginia
18. ---------- United Kingdom
19. Ross Tew ----------