Question
What is the area of the largest rectangular field which can be enclosed with $$200\,m$$ of fencing ?
A.
$$1600\,{m^2}$$
B.
$$2100\,{m^2}$$
C.
$$2400\,{m^2}$$
D.
$$2500\,{m^2}$$
Answer :
$$2500\,{m^2}$$
Solution :
Let length and breadth of rectangular field be $$x$$ and $$y$$ respectively
$$\therefore \,2\left( {x + y} \right) = 200 \Rightarrow y = 100 - x$$
and area $$A = xy = x\left( {100 - x} \right)$$
$$\because \,\frac{{dA}}{{dx}} = 100 - 2x$$
Put $$\frac{{dA}}{{dx}} = 0$$ for maxima or minima
$$\eqalign{
& 100 - 2x = 0 \cr
& \Rightarrow x = 50\,\,\, \Rightarrow y = 50 \cr} $$
Now, $$\frac{{{d^2}A}}{{d{x^2}}} = - 2 < 0,$$ which shows maximum, independent of values of $$x$$ and $$y,$$ but only when they are equal.
$$\therefore \,A$$ is maximum at $$x = 50$$
Hence, required area $$ = 50\left( {100 - 50} \right) = 50 \times 50 = 2500\,{m^2}$$