Question

Farmer Ed has 9,500 meters of fencing, and wants to enclose a rectangular plot that boarders on a river. If Farmer Ed does not fence the side along the river, what is the largest area that can be enclosed?The largest area that can be enclosed is____square meters

Answer 1

`Area_(max)=11,281,250m^(2)`

Step-by-step explanation:

We were given the following information:

Ed has 9,500 meters of fencing

Ed does not fence the side along the river

`\begin{gathered} Perimeter=9,500m \\ Perimeter=length+2*width \\ length=9,500-2x \\ width=x \\ The\text{ area of a rectangle is given by:} \\ Area=length*width \\ Area=\lparen9,500-2x)*x \\ Area=9,500x-2x^2 \\ Area=-2x^2+9,500x \\ The\text{ maximum point occurs at the vertex \lparen to obtain the largest area that can be enclosed\rparen:} \\ x=-(b)/(2a)=-(9,500)/(2\left(-2\right)) \\ x=(9,500)/(4)=2,375 \\ x=2,375 \\ Substitute\text{ this into the formula for ''length'', we have:} \\ length=9,500-2\left(2,375\right) \\ length=9,500-4,750=4,750 \\ length=4,750m \\ w\imaginaryI dth=x=2,375m \\ Area_(max)=length*width \\ Area_(max)=4,750*2,375 \\ Area_(max)=11,281,250m^2 \\ \\ \therefore Area_(max)=11,281,250m^2 \end{gathered}`

Therefore, the largest area that can be enclosed is 11,281,250 sq. meters