Question

A farmer has 500 yards of fencing with which to enclose a rectangular paddock. What is the largest area that can be enclosed?

Answer 1

Given:

The farmer has 500 yards of fencing with which to enclose a rectangular paddock.

Let x be the width of the rectangle and l be the length of the rectangle.

The equation for the total fencing available is given by,

`\begin{gathered} 2x+2l=500 \\ \text{Simplify} \\ x+l=250 \\ l=250-x \end{gathered}`

So, the area is given as,

`\begin{gathered} A=x\cdot l \\ A=x\cdot(250-x) \\ A=250x-x^2 \end{gathered}`

Find the critical values,

`\begin{gathered} A^(\prime)=(d)/(dx)(250x-x^2) \\ A^(\prime)=250-2x \\ \text{Set A'=0} \\ 250-2x=0 \\ x=125 \end{gathered}`

Test critical value,

`A^(\doubleprime)=(d^2)/(dx^2)(250x-x^2)=-2<0\text{ for all x, }\Rightarrow x=125\text{ is maximum}`

So, the maximum area is,

`A=250x-x^2=250(125)-125^2=15625\text{ square yards.}`

Answer: the maximum area is 15625 square yards.