Question

You have D dollars to buy fence to enclose a rectangular plot of land (see figure at right). The fence for the top and bottom costs $4 per foot and for the sides it costs $3 per foot . Find the dimensions of the plot with the largest area. For this largest plot , how much money was used for the top and bottom, x, and for the sides, ?

Answer 1

The perimeter of the rectangular plot of land is given by the expression below

`P=2x+2y`

On the other hand, since the available money to buy fence is D dollars,

`\begin{gathered} D=4(2x)+3(2y) \\ \Rightarrow D=8x+6y \\ D\rightarrow\text{ constant} \end{gathered}`

Furthermore, the area of the enclosed land is given by

`A=xy`

Solving the second equation for x,

`\begin{gathered} D=8x+6y \\ \Rightarrow x=(D-6y)/(8) \end{gathered}`

Substituting into the equation for the area,

`\begin{gathered} A=((D-6y)/(8))y \\ \Rightarrow A=(D)/(8)y-(3)/(4)y^2 \end{gathered}`

To find the maximum possible area, solve A'(y)=0, as shown below

`\begin{gathered} A^(\prime)(y)=0 \\ \Rightarrow(D)/(8)-(3)/(2)y=0 \\ \Rightarrow(3)/(2)y=(D)/(8) \\ \Rightarrow y=(D)/(12) \end{gathered}`

Therefore, the corresponding value of x is

`\begin{gathered} y=(D)/(12) \\ \Rightarrow x=(D-6((D)/(12)))/(8)=(D-(D)/(2))/(8)=(D)/(16) \end{gathered}`

Thus, the dimensions of the fence that maximize the area are x=D/16 and y=D/12.

As for the used money,

`\begin{gathered} top,bottom:(8D)/(16)=(D)/(2) \\ Sides:(6D)/(12)=(D)/(2) \end{gathered}`

Half the money was used for the top and the bottom, while the other half was used for the sides.