32.8k views
4 votes
A farmer has 500 yards of fencing with which to enclose a rectangular paddock. What is the largest area that can be enclosed?

1 Answer

2 votes

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.

User Shulito
by
8.4k points
Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.

9.4m questions

12.2m answers

Categories