Hello, 0017024651!
A farmer has 3,000 feet of fence to enclose a rectangular field
and subdivide it into 3 rectangular plots.
If $\displaystyle x$ denotes the width of the field and $\displaystyle y$ the length,
find the value of $\displaystyle x$ so that the total area of the field is maximized. Code:
*-------*-------*-------*
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x| x| x| x|
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*-------*-------*-------*
: - - - - - y - - - - - :
He has four fences which are $\displaystyle x$ feet long
. . and two fences which are $\displaystyle y$ feet long.
So total fencing is: .$\displaystyle 4x + 2y \:=\:3000\quad\Rightarrow\quad y \:=\:1500-2x$ .[1]
The area of the field is: .$\displaystyle A \:=\:xy$ .[2]
Substitute [1] into [2]: .$\displaystyle A \;=\;x(1500-2x) \quad\Rightarrow\quad A \;=\;1500x-2x^2$
And that is the function we must maximize . . .