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Fudgin [204]
10 months ago
6

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 co

sts $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, ?

Mathematics
1 answer:
alex41 [277]10 months ago
7 0

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=\frac{D-6y}{8} \end{gathered}

Substituting into the equation for the area,

\begin{gathered} A=(\frac{D-6y}{8})y \\ \Rightarrow A=\frac{D}{8}y-\frac{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\frac{D}{8}-\frac{3}{2}y=0 \\ \Rightarrow\frac{3}{2}y=\frac{D}{8} \\ \Rightarrow y=\frac{D}{12} \end{gathered}

Therefore, the corresponding value of x is

\begin{gathered} y=\frac{D}{12} \\ \Rightarrow x=\frac{D-6(\frac{D}{12})}{8}=\frac{D-\frac{D}{2}}{8}=\frac{D}{16} \end{gathered}<h2>Thus, the dimensions of the fence that maximize the area are x=D/16 and y=D/12.</h2><h2>As for the used money,</h2>\begin{gathered} top,bottom:\frac{8D}{16}=\frac{D}{2} \\ Sides:\frac{6D}{12}=\frac{D}{2} \end{gathered}<h2>Half the money was used for the top and the bottom, while the other half was used for the sides.</h2>

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