Question

A farmer has a straight, fenced road along the boundary of his property. He wishes to fence an enclosure with the largest possible area and has enough materials to erect 1000m of fence. What would be the dimensions of the enclosure, assuming that he uses the existing boundary fence as one of the sidesI’m studying quadratic equations at the moment and I’m struggling with this question

Answer 1

Answer: $\begin{gathered} Largest\text{ possible area = 125000 m}^2 \\ \\ The\text{ dimension which will give the largest posible area:} \\ length\text{ = 500 m, width = 250 m} \end{gathered}$

Explanation:

The total length of the fence = 1000m

Assumption: The farmer uses the existing boundary fence as one of the sides

This means we will be considering 3 sides

Total length = 2 widths + 1 length

let the width of the fence = w

2 widths = 2(x) = 2w

Total length = 2w + length

1000 = 2w + length

length = 1000 - 2w

To solve the question, we will make an illustration of the fencing

Area of the fence = length × width

$\begin{gathered} \text{A = \lparen1000 - 2w\rparen}\times\text{ w} \\ \text{A = 1000w - 2w}^2 \end{gathered}$

To get the largest possible area, we will find the derivative with respect to w:

$\begin{gathered} \frac{dA}{dw}\text{ = }\frac{d(1000w\text{ - 2w}^2)}{dw} \\ A^{\prime}\text{ = 1000 - 4w} \\ We\text{ will equate the derivative to zero:} \\ 0\text{ = 1000 - 4w} \\ 4w\text{ = 1000} \\ \\ divide\text{ both sides by 4:} \\ w\text{ = 1000/4} \\ w\text{ = 250} \end{gathered}$

To get the length, we will substitute the value of the width into the length formula:

$\begin{gathered} \text{length = 1000 - 2w } \\ length\text{ = 1000 - 2\lparen250\rparen} \\ length\text{ = 500 m} \\ \\ width\text{ = 250 m} \end{gathered}$

The largest possible area:

$\begin{gathered} Area\text{ = l }\times w\text{ = 500 }\times\text{ 250} \\ Largest\text{ possible area = 125000 m}^2 \\ \\ The\text{ dimension which will give the largest posible area:} \\ length\text{ = 500 m, width = 250 m} \end{gathered}$