Let
x-----> the length of rectangle
y-----> the width of rectangle
we know that
perimeter of rectangle=2*[x+y]
perimeter of rectangle=82 m
82=2*[x+y]---> divide by 2 both sides---> 41=x+y--> y=41-x---> equation 1
Area of rectangle=x*y
substitute equation 1 in the area formula
Area=x*[41-x]----> 41x-x²
using a graph tool
see the attached figure
the vertex is the point (20.5,420.25)
that means
for x=20.5 m ( length of rectangle)
the area is 420.25 m²
y=420.25/20.5----> 20.5 m
the dimensions are
20.5 m x 20.5 m------> is a square
the answer part 1) <span>
the dimensions of the rectangular with Maximum area is a square with length side 20.5 meters</span>
Part 2)<span>b) Suppose 41 barriers each 2m long, are used instead. Can the same area be enclosed?
</span>divide the length side of the square by 2
so
20.5/2=10.25--------> 10 barriers
the dimensions are 10 barriers x 10 barriers
10 barriers=10*2---> 20 m
the area enclosed with barriers is =20*20----> 400 m²
400 m² < 420.25 m²
so
the answer Part 2) is
<span>the area enclosed by the barriers is less than the area enclosed by the rope
</span>
Part 3)<span>How much more area can be enclosed if the rope is used instead of the barriers
</span>
area using the rope=420.25 m²
area using the barriers=400 m²
420.25-400=20.25 m²
the answer part 3) is20.25 m²
