Has 320 yards of fencing to enclose a rectangular area. find the dimensions of the rectangle that maximize the enclosed area. what is the maximum area

Let the dimensions of the rectangular area be a × b.

The perimeter of the rectangular area is 320 yards, 

so

2a+2b=320

dividing by 2: 

a+b=160

b=160-a

So the width and the length of the rectangular area are a and 160-a.

The area of the rectangle is a function depending on a :



we can see that A is a quadratic function, so its graph is a parabola. The parabola opens downwards because if we write A(a) in the general form       -a^2+160a, we see that the coefficient of a^2 is negative.

Thus the maximal value the function can take is the y-coordinate of the vertex.

, so the roots are easily determined as 0 and 160.

the midpoint of 0 and 160 is 80.

the vertex is (80, A(80)) =(80, 80*80)=(80, 6400)

6400 square yard.