Answer
a) 40 cm
b) $20-w = l$
c) $A = -w^2+20w$
d) Please see the graph.
e) (10,100) This means that the maximum area of the rectangle is 100 $cm^2$
Work Step by Step
a) I started with a 40 cm string.
b)
$perimeter = 2*w+2*l$
$P = 2w+2l$
$40 = 2w+2l$
$40/2 = (2w+2l)/2$
$20 = w+l$
$20-w = w+l-w$
$20-w = l$
c)
$area = l*w$
$area = (20-w)*w$
$A = 20w-w^2$
$A = -w^2+20w$
d) The graph was made using graphing software.
Vertex:
$x=-b/2a$
$x = -20/2*-1$
$x = -20/-2 = 10$
$A = -(10)^2+20*10$
$A = -100+200 = 100$
Axis of symmetry is $x=10$ (per the work above for finding the vertex)
One other point on the curve:
$x=0$
$A = -(0)^2+20*0$
$A = -0 +0 = 0$
e)
Per part d), the vertex is (10,100).
