Problem solving?!

Question

What is the area of the largest rectangle with a perimeter of 160?

Answer 1

The largest area of a rectangle is always a square.

P=160

160/4=40 units

The rectangle with the largest area is the square whose sides all measure 40 units.

A=40*40

A=1600 square units

Look at it this way...

P=2l+2w

160=2l+2w

80=l+w

w=width and length=l, and l=80-w

Area=length*width

A=w(80-w)

A=80w-w²

the maximum area occurs when w=-b/2a where a=-1 and b=80 in the quadratic A=80w-w²

w=(-80)/(2*[-1])

w=(-80)/(-2)

w=40 units, thus giving a square

 

Answer 2

The largest area a rectangle can hold is in the shape of a square, so

p = 4s

160 units = 4s

40 units = s

So each side is 40 units long.

 

The area of a square equals the side squared, so

a = 40 units * 40 units

a = 1600 units²

 

So the area is 1600 units².