Question

A campground owner plans to enclose a rectangular field adjacent to a river. The owner wants the field to contain $125,000$ square meters. No fencing is required along the river. What dimensions (in meters) will use the least amount of fencing?

Answer 1

Step-1: Finding the length of the fencing:

Given area of rectangular field is $125,000{\mathrm{m}}^2$
Area of rectangle $=L\times W$, where $L=$Length and $W=$Width
$$\begin{gathered} \Rightarrow LW=125000 \\ \Rightarrow W=\frac{125000}{L} \end{gathered}$$
Length of the fencing $=L+2W$ (Since no fencing is required along the river )
$=L+\frac{250000}{L}$
Step-2: Calculating the value of length and width:

For maximizing ,$\frac{\mathrm{d}\left(\mathrm{perimeter}\right)}{\mathrm{dL}}=0$
$$\begin{gathered} \Rightarrow 1-\frac{250000}{L^2}=0 \\ \Rightarrow L^2=250000 \\ \Rightarrow L=\sqrt{250000} \\ =500\mathrm{m}\left[\mathrm{double}\mathrm{derivative}\mathrm{will}\mathrm{be}\mathrm{positive}\mathrm{for}\mathrm{positive}\mathrm{value}\right] \end{gathered}$$
$$\begin{gathered} \Rightarrow W=\frac{125000}{500} \\ =250\mathrm{m} \end{gathered}$$

Hence, dimensions will use the least amount of fencing is $500\mathrm{m}\&250\mathrm{m}$.