Diana has available 80 yards of fencing and wishes to enclose a rectangular area. (a) express the area a of the rectangle as a function of the width w of the rectangle. (b) for what value of w is the area largest? (c) what is the maximum area?

a) A = 40*W – W^2

b) W = 20

c ) A = 400

Step-by-step explanation:

a)Let be

W = width W of the rectangle

L = lenght of the rectangle

P = Perimeter of the rectangle

A = area of the rectangle  

P = 2*L + 2*W

80 = 2*L + 2*W

So, 2*L = 80 – 2*W

L = 40 – W

A = L * W

Replacing L

A = (40 – W)*W

A = 40*W – W^2

b)To find the máximum value for W, we derivate area and equal to zero

A’ = 40 – 2*W

40 – 2*W = 0

2*W = 40

W= 20

c)With the value for W, we find L

L = 40 – W

L = 40 – 20

L = 20

A = W*L

A = 20 * 20

A = 400