Question

Suppose that you have 840 feet of rope and want to use it to make a rectangle. What dimensions should you make your rectangle if you want to enclose the maximum possible 10 area?

Answer 1

Let length of rectangle is x .

we know

perimeter = 2(length+breadth)

perimeter should be length of rope 840

2(length+breadth)= 840

length +breadth=420

breadth =420-length = 420-x

we know. Area= length *breadth

so

area=x (420-x) = 420x-x²

Area= -x²+420x

^{This is a quadratic function.}
A quadratic function of the form ax²+bx+c=0 , has maximum when a is negative and maximum occur at

$\large x=\frac{-b}{2a}$

for our area equation a= -1 ,b=420

so maximum is when

$\large x=\frac{-420}{2*-1}=210$

So length is 210 and breadth is 210 when it enclose maximum area

So rectangle should be a square of side 210 to have maximum area .