Optimization problems #1

Hello, jsu03!

Quote:

1. A farmer with 750 ft of fencing wants to enclose a rectangular area
and then divide it into four pens with fencing parallel to one side of the rectangle.
What is the largest possible total area of the four pens?

First, you need to visualize the problem . . . carefully.

Code:

: - - - - - x - - - - - : *-----*-----*-----*-----* | | | | | | | | | | y| y| |y |y |y | | | | | | | | | | *-----*-----*-----*-----* : - - - - - x - - - - - :

The region is a rectangle with length $x$ and width $y.$

It is divided into 4 pens with three more fences of width $y.$

The fencing is limited to 750 feet: . $2x+ 5y \:=\:750 \quad\Rightarrow\quad y \:=\:150 - \tfrac{2}{5}x$ .[1]

The area of the rectangle is: . $A \;=\;xy$ .[2]

Substitute [1] into [2]: . $A \;=\;x\left(150 - \tfrac{2}{5}x\right) \quad\Rightarrow\quad A \;=\;150x - \tfrac{2}{5}x^2$

Maximize: . $A' \:=\:150-\tfrac{4}{5}x\:=\:0 \quad\Rightarrow\quad\boxed{ x \:=\:\tfrac{375}{2}}$

Substitute into [1]: . $y \;=\;150-\tfrac{2}{5}\left(\tfrac{375}{2}\right) \quad\Rightarrow\quad\boxed{ y \:=\:75}$

The maximum area is: . $A \;=\;\left(\tfrac{375}{2}\right)(75) \;=\;\boxed{14,\!062.5\text{ ft}^2}$